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Mechanics of jet propulsion in a hydromedusean jellyfish DeMont, Malcolm Edwin 1986

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MECHANICS OF JET PROPULSION IN A HYDROMEDUSEAN JELLYFISH by MALCOLM EDWIN DEMONT B.Sc. , U n i v e r s i t y of King's C o l l e g e and Dal h o u s i e U n i v e r s i t y , 1979 M . S c , Da l h o u s i e U n i v e r s i t y , 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Zoology We accept t h i s t h e s i s as conforming to the r e a u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1986 © Malcolm Edwin DeMont, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 i i ABSTRACT A n o n - d e s t r u c t i v e t e s t was developed to measure the s t a t i c mechanical p r o p e r t i e s of the locomotor s t r u c t u r e ( b e l l ) i n the hydromedusean j e l l y f i s h , P o l y o r c h i s p e n i c i l l a t u s . A n o n l i n e a r s t r e s s - s t r a i n r e l a t i o n s h i p was found, and the mean s t a t i c -2 s t r u c t u r a l s t i f f n e s s of the b e l l was 150 N m . Observations that showed the n a t u r a l changes i n the geometry of the b e l l d u r i n g deformation were used to estimate the s t a t i c modulus of e l a s t i c i t y of the mesogleal m a t e r i a l , and gave a modulus of -2 400 N m . Dynamic measurements on i s o l a t e d samples of -i mesoglea gave a mean storage modulus of 1000 N m . The r e s i l e n c e of the m a t e r i a l was about 58%. These data were i n t e g r a t e d to i n f e r t h a t the dynamic s t r u c t u r a l s t i f f n e s s of T — the b e l l i s at l e a s t 400 N m . Attempts to q u a n t a t i t i v e l y measure the dvnamic s t r u c t u r a l s t i f f n e s s imply that the dynamic -2 -2 s t r u c t u r a l s t i f f n e s s must l i e between 400 N m and 1000 N m A l l , or most, of the p o t e n t i a l energy s t o r e d i n the mesoglea d u r i n g c o n t r a c t i o n s of the b e l l i s a p p a r e n t l y s t o r e d as s t r a i n energy i n the r a d i a l mesogleal f i b e r s . The mechanical energy generated by the c o n t r a c t i o n of the subumbrellar swimming muscles to power the j e t c y c l e was measured. T h i s energy was e x p e r i m e n t a l l y p a r t i t i o n e d i n t o three components d u r i n g the c o n t r a c t i o n . The a l g e b r a i c sum of these components was taken t o be the mechanical energy i i i crenerated by the muscles d u r i n g the i e t c y c l e , and was between -5 -4 8.9 x 10 and 1.4 x 10 J . Energy from one of these components i s s t o r e d as s t r a i n energy i n the mesoglea and powers the r e f i l l i n g phase. The mesoglea can c l e a r l y a c t as an e f f e c t i v e e l a s t i c s t r u c t u r e to completely antagonize the c o n t r a c t i o n of the swimming muscles, and may be designed to f u n c t i o n at some optimum. The mechanical s i g n i f i c a n c e of e l a s t i c energy storage systems i n j e t - p r o p e l l e d animals i s d i s c u s s e d , and t h i s s i g n i f i c a n c e i s c l e a r l y d i s p l a y e d i n P o l v o r c h i s . The unusual long d u r a t i o n a c t i o n p o t e n t i a l of the swimming muscles may have some mechanical s i g n i f i c a n c e w i t h regard to the e l a s t i c energy storage system. I t i s suggested that the a c t i o n p o t e n t i a l of v e r t e b r a t e c a r d i a c muscle may have the same mechanical f u n c t i o n . The locomotor b e l l of the hydromedusean j e l l y f i s h was modelled as a h a r m o n i c a l l y f o r c e d damped o s c i l l a t o r . The robustness of the model was t e s t e d and v e r i f i e d by comparing estimates of the work done d u r i n g the c o n t r a c t i o n phase p r e d i c t e d by the model to analogous v a l u e s measured i n completely independent experiments. Data suggest that the animals swim at a frequency t h a t i s at or near the resonant frequency of the locomotor apparatus. The i m p l i c a t i o n s of t h i s phenomenon to the mechanics and p h y s i o l o g y of the system are d i s c u s s e d . I f the swimming muscles f o r c e the b e l l at i t s resonant frequency, as opposed to a s i n g l e c o n t r a c t i o n at the i v same r a t e of deformation, the amplitude of the o s c i l l a t i o n w i l l be i n c r e a s e d by about 40%, and the e n e r g e t i c requirement f o r the c y c l e w i l l be reduced by about 24% to 37% of the t o t a l c ost of the c y c l e . The advantages of f o r c i n g the s t r u c t u r e at i t s resonant frequency seem q u i t e remarkable. Two aspects of the p h y s i o l o g y of the swimming muscles are probably f u n c t i o n a l l y r e l a t e d to t h i s phenomenon and are d i s c u s s e d . V TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS v LIST OF TABLES J v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS i x I. INTRODUCTION A. JET-PROPELLED LOCOMOTION 1 B. ANATOMY OF THE BELL 8 I I . MECHANICAL PROPERTIES OF THE LOCOMOTOR STRUCTURE A. INTRODUCTION 12 B. MATERIALS AND METHODS 14 1. S t a t i c T e s t s 14 2. Dynamic Mechanical T e s t s 20 a. I s o l a t e d Mesoglea 20 b. I n t a c t Locomotor S t r u c t u r e .... 23 3. Observations of Mesogleal Geometry .... 26 4. S t a t i c t i c s 27 C. RESULTS 28 1. S t a t i c T e s t s 28 2. Dynamic T e s t s on I s o l a t e d Mesoglea .... 34 3. Observations of Mesogleal Geometry .... 39 4. Dynamic T e s t s on the I n t a c t B e l l 48 D. DISCUSSION 50 v i I I I . ENERGETICS OF THE JET CYCLE A. INTRODUCTION 57 B. MATERIALS AND METHODS 60 C. RESULTS 64 1. The C o n t r a c t i o n Phase 64 2. The R e f i l l i n g Phase 75 D. DISCUSSION 77 IV. THE PRESENCE AND IMPORTANCE OF A RESONANT PHENOMENON IN THE LOCOMOTOR STRUCTURE A. INTRODUCTION 87 B. MATERIALS AND METHODS 90 1. Experimental 90 2. A n a l y t i c a l 91 C. RESULTS 97 D. DISCUSSION 110 V. SUMMARY 124 REFERENCES 125 APPENDIX I 132 APPENDIX I I 136 APPENDIX I I I 138 v i i L I S T O F T A B L E S T a b l e 2 . 1 M e c h a n i c a l t e s t d a t a 3 5 T a b l e 3 . 1 E n e r g i e s o f t h e jet c y c l e 7 0 T a b l e 4 . 1 P a r a m e t e r s o f t h e o s c i l l a t o r 1 0 2 v i i i LIST OF FIGURES F i g u r e 1.1 Anatomy of the b e l l 11 F i g u r e 2.1 Schematic of apparatus f o r s t a t i c t e s t s .... 16 F i g u r e 2.2 Pressure-volume curves f o r the locomotor b e l l 30 F i g u r e 2.3 S t r e s s - s t r a i n curves f o r the locomotor b e l l 33 F i g u r e 2.4 Dynamic mechanical data of i s o l a t e d mesoglea 38 F i g u r e 2.5 Morphology of the deformation 42 F i g u r e 2.6 R e l a t i o n of i n t e r r a d i a l to t o t a l s t r a i n .... 44 F i g u r e 2.7 R e l a t i o n of r a d i a l s t r a i n and c i r c u m f e r e n t i a l s t r a i n 47 F i g u r e 3.1 T y p i c a l pressure-diameter data f o r n a t u r a l c o n t r a c t i o n s 66 F i g u r e 3.2 Pressure-volume loop f o r n a t u r a l c o n t r a c t i o n 68 F i g u r e 3.3 I n e r t i a l pressure generated d u r i n g a n a t u r a l c o n t r a c t i o n 74 F i g u r e 4.1 Free damped v i b r a t i o n of the b e l l 99 F i g u r e 4.2 Resonance curve of the b e l l 105 F i g u r e 4.3 T y p i c a l t r a i n s of c o n t r a c t i o n s of the b e l l 109 i x ACKNOWLEDGEMENTS My time spent here at UBC has le a d to two d i s c o v e r i e s . The importance of the f i r s t w i l l be e v a l u a t e d by those r e a d i n g the pages that f o l l o w . The second i s unquestionably g r e a t . I d i s c o v e r e d how p r i v i l e g e d I was, so long ago, when John G o s l i n e allowed me to become a member of h i s l a b . I wish to thank him, and a l l the members of my s u p e r v i s o r y committee f o r t h e i r a s s i s t a n c e throughout t h i s work. A s p e c i a l thanks t o Bob Blake and B i l l Milsom. I a l s o wish to thank the s t a f f of the Bamfield Marine S t a t i o n f o r t h e i r a s s i s t a n c e i n my f i e l d work. 1 CHAPTER I. INTRODUCTION A. JET-PROPELLED LOCOMOTION Aqua t i c organisms use a v a r i e t y of mechanisms to swim. In a l l of these modes, animals expend energy to move v a r i o u s p a r t s of t h e i r bodies to do work on t h e i r environment, and thus generate the t h r u s t that a c t u a l l y p r o p e l s the animal. Some animals use a j e t r e a c t i o n mechanism to p r o p e l themselves. These animals i n c l u d e cephalopod m o l l u s c s , d r a g o n f l y nymphs, s a l p s , siphonophores, s c a l l o p s and medusae (Hughes, 1958; Packard, 1969; Moore and Trueman, 1971; G l a d f e l t e r , 1972; Johnson, Soden and Trueman, 1973; Bone and Trueman, 1982 and 1983; D a n i e l , 1983, 1984, 1985; G o s l i n e and Shadwick, 1983; G o s l i n e et a l . , 1983; G o s l i n e and DeMont, 1985). T h i s work pro v i d e s a d e t a i l e d account of the mechanics of the j e t c y c l e i n the hydromedusean j e l l y f i s h P o l y o r c h i s p e n i c i l l a t u s . In j e t p r o p u l s i o n , animals e j e c t a volume of water enclosed by p o r t i o n s of t h e i r body. The water e x i t s through an o r i f i c e at one end of the animal. By simple c o n s e r v a t i o n of momentum p r i n c i p l e s , the animal i s p r o p e l l e d i n the d i r e c t i o n o p p o s i t e the j e t c r e a t e d by the e j e c t i o n of the water. T h i s e j e c t i o n i s powered by the c o n t r a c t i o n of swimming muscles that deform the p o r t i o n of the body e n c l o s i n g the water. R e f i l l i n g of the c a v i t y p r e v i o u s l y c o n t a i n i n g the water i s g e n e r a l l y a 2 p a s s i v e p r o c e s s , powered by p o t e n t i a l energy s t o r e d e l a s t i c a l l y i n the deformed s t r u c t u r e . Thus the e l a s t i c i t y of the whole locomotor s t r u c t u r e p l a y s an important r o l e i n the mechanics of the j e t c y c l e . The presence and importance of e l a s t i c m a t e r i a l i n the locomotor t i s s u e of another j e t - p r o p e l l e d animal, the s q u i d , has a l r e a d y been examined i n some d e t a i l ( G o s l i n e et al_. , 1982; G o s l i n e and Shadwick, 1983 and G o s l i n e and DeMont, 1985). The present study was i n i t i a t e d by an i n t e r e s t i n comparing the mechanism of t h r u s t g e n e r a t i o n i n a s q u i d to t h a t i n a j e l l y f i s h . Both organisms are j e t - p r o p e l l e d , but d i f f e r i n s e v e r a l important c h a r a c t e r s . The s q u i d i s a very a c t i v e predator and n e c e s s a r i l y needs a w e l l developed locomotor system to reach the h i g h speeds needed to capture i t s prey. The j e l l y f i s h i s not an a c t i v e p r e d a t o r , and needs i t s locomotor system only to maintain i t s e l f i n the water column. Thus i t s locomotor system does not need to be as developed as the locomotor system i n the squid. A d e t a i l e d comparsion of the locomotor systems i s made i n Chapter I I I , but i t i s worthwhile to mention major d i f f e r e n c e s here. The locomotor system of squid c o n t a i n s two sets of muscles, and each set i s a p p a r e n t l y capable of powering p o r t i o n s of the j e t c y c l e ( G o s l i n e et a l . , 1982). T i s s u e e l a s t i c i t y can antagonize e i t h e r of the s e t s of muscles. The locomotor system of j e l l y f i s h c o n t a i n s only one set of swimming muscles, and these 3 are antagonized completely by the e l a s t i c i t y of the t i s s u e comprising the locomotor s t r u c t u r e . The mechanical p r o p e r t i e s of the locomotor s t r u c t u r e important to the swimming e n e r g e t i c s have not been examined e x t e n s i v e l y i n any j e l l y f i s h . P r e l i m i n a r y work, however, has been completed on the mechanical p r o p e r t i e s of the locomotor s t r u c t u r e s i n some s p e c i e s . For i n s t a n c e , Bone and Trueman (1982) examined j e t p r o p u l s i o n i n siphonophores, but were unable to measure the e l a s t i c r e s t o r i n g f o r c e of the locomotor t i s s u e because they could not simulate n a t u r a l deformations of the b e l l . D a n i e l (1983) modelled the mechanics of medusean j e t p r o p u l s i o n and w i t h some assumptions on the mechancial p r o p e r t i e s of the locomotor s t r u c t u r e , found that a decrease i n the e l a s t i c energy storage c a p a b i l i t i e s of the s t r u c t u r e g i v e s r i s e to a decrease i n the o v e r a l l locomotor e f f i c i e n c y . A major p o r t i o n of the work presented i n t h i s t h e s i s i s devoted to the f i r s t measurements of p h y s i o l o g i c a l l y important mechanical p r o p e r t i e s of the e l a s t i c locomotor s t r u c t u r e . I t i s i n t e r e s t i n g that any e l a s t i c s t r u c t u r e d r i v e n by some p e r i o d i c f o r c e shows a response i n the amplitude of the r e s u l t a n t d r i v e n o s c i l l a t i o n t h a t i s dependent on the frequency of the. d r i v i n g f o r c e . At a s p e c i f i c frequency, c a l l e d the resonant frequency of the o s c i l l a t o r , the amplitude of the o s c i l l a t i o n i s maximized as compared to amplitudes of 4 o s c i l l a t i o n when f o r c e d a t f r e q u e n c i e s above or below the resonant frequency. B i o l o g i c a l s t r u c t u r e s designed to minimize the cost of l a r g e amplitude o s c i l l a t i o n s might take advantage of resonance simply by having the muscles d r i v e the s t r u c t u r e a t i t s resonant frequency. T h i s i d e a i s not new. I t has been proposed t h a t locomotor systems i n t e r r e s t r i a l animals are being f o r c e d t o work at the resonant frequency of the locomotor s t r u c t u r e (McMahon, 1975, 1985; T a y l o r , 1985). Some animals (dogs, c a t s , r a b b i t s , sheep, c a t t l e ) pant a t a frequency that i s near the resonant frequency of the r e s p i r a t o r y system (see f o r example Agostoni et al., 1970). I propose that when the hydromedusean j e l l y f i s h P o l y o r c h i s i s swimming i n continuous bouts of c o n t r a c t i o n s , the swimming muscles are f o r c i n g the b e l l , or locomotor s t r u c t u r e , to o s c i l l a t e at i t s resonant frequency. The hypothesis of r e s o n a t i n g locomotor s t r u c t u r e s has not been suggested p r e v i o u s l y f o r any a q u a t i c organism. A summary of the work used i n the present study to t e s t the hypothesis f o l l o w s . In Chapter I I , the methods t h a t were developed to measure the s t a t i c s t r u c t u r a l s t i f f n e s s of the locomotor s t r u c t u r e are d e s c r i b e d . These methods u l t i m a t e l y p r o v i d e d a measure of the dynamic s t r u c t u r a l s t i f f n e s s of the b e l l . Two major problems c r e a t e d d i f f i c u l t i e s i n making these measurements. F i r s t l y , the s t a t i c f o r c e s r e q u i r e d t o deform the t i s s u e are very small and apparatus r e q u i r e d to simulate n a t u r a l c o n t r a c t i o n s were 5 d i f f i c u l t to c o n s t r u c t . T h i s i s ev i d e n t from the l a c k of any success by other workers. The magnitude of these f o r c e s can be emphasized through comparisons of the measured s t a t i c modulus of e l a s t i c i t y of the mesogleal m a t e r i a l comprising the b e l l to other m a t e r i a l s , both man-made and n a t u r a l . The mean s t a t i c modulus of e l a s t i c i t y of the b e l l mesoglea i s shown i n Chapter 2 -2 II to be about 4 x 10 N m . For comparison, the modulus of 4 -2 mesoglea of sea anemones i s about 1 x 10 N m (Gosline,' 1971), the s h e l l membrane of egg has a modulus of about 6 - 2 7 - 2 8 x 10 N m , human c a r t i l a g e about 2.4 x 10 N m , bone 10 -2 about 2.1 x 10 N m , o r d i n a r y g l a s s e s about 10 -2 12 -2 7.0 x 10 N m , and diamond about 1.2 x 10 N m (Gordon, 1978, p. 54). These comparisons are r a t h e r a r b i t r a r y , but i t i s obvious t h a t j e l l y f i s h mesoglea has a modulus that i s orders of magnitude s m a l l e r than other ' s o f t ' m a t e r i a l s . One problem wi t h these comparisons i s that many m a t e r i a l s e x h i b i t frequency dependent p r o p e r t i e s , and i t i s very d i f f i c u l t to c h a r a c t e r i z e the mechanical p r o p e r t i e s by a s i n g l e v a l u e , s i n c e the r a t e of deformation of the m a t e r i a l d u r i n g the t e s t i n g determines the magnitude of the s t i f f n e s s . That leads d i r e c t l y i n t o the second problem w i t h the measurements made i n Chapter I I . The mesogleal m a t e r i a l that comprises the s t r u c t u r e has frequency dependent p r o p e r t i e s . S t a t i c t e s t s were comparatively e a s i l y made, but the r e s u l t s p r ovide l i t t l e i n f o r m a t i o n t o an understanding of the s t r u c t u r a l p r o p e r t i e s 6 t h a t e x i s t a t n a t u r a l r a t e s of deformation. I t was not p o s s i b l e to measure a s i n g l e v alue f o r the dynamic s t r u c t u r a l s t i f f n e s s a t the n a t u r a l r a t e of deformation. However, i t was p o s s i b l e to set a range of valu e s f o r the s t i f f n e s s , and data -2 show that the dynamic s t r u c t u r a l s t i f f n e s s i s between 400 N m -2 and 1000 N m . T h i s value has p h y s i o l o g i c a l importance, s i n c e i t can be used to estimate the amount of energy the swimming muscles must generate t o deform the t i s s u e , and t h i s i n f o r m a t i o n i s r e q u i r e d f o r the an a l y s e s completed i n Chapters I I I and IV. Most of the measurments made i n Chapters I I I and IV use, i n some way, the dynamic s t r u c t u r a l s t i f f n e s s measured i n Chapter I I . Most of the measurements, t h e r e f o r e , cannot be presented as a unique q u a n t i t y , but are expressed as a range, r e f l e c t i n g the range n e c e s s a r i l y set f o r the dynamic s t r u c t u r a l s t i f f n e s s measured i n Chapter I I . The l a s t two Chapters are very much i n t e r c o n n e c t e d . In Chapter I I I , the energy generated d u r i n g the j e t c y c l e i s e x p e r i m e n t a l l y measured through a d e t a i l e d a n a l y s i s of a s e r i e s of c o n t r a c t i o n s f o r a s i n g l e animal. The energie s are measured s e p a r a t e l y , and summed i n the end to g i v e the t o t a l mechanical energy generated d u r i n g the c o n t r a c t i o n . Data from Chapter II are used to estimate the q u a n t i t y of energy r e q u i r e d to deform the t i s s u e . The e n e r g e t i c s of the r e f i l l i n g phase i s a l s o a nalyzed, and thus the e n e r g e t i c s of the e n t i r e j e t c y c l e i s presented. 7 In Chapter IV, the locomotor s t r u c t u r e i s modelled as a h a r m o n i c a l l y f o r c e d damped o s c i l l a t o r . Three parameters are needed to c h a r a c t e r i z e an o s c i l l a t o r ; (1) i t s damping c o e f f i c i e n t , (2) i t s s p r i n g c o n s t a n t , and (3) i t s mass. The damping parameters of the b e l l were measured from some r a r e data on f r e e damped o s c i l l a t i o n s , f o r c e d by a s i n g l e c o n t r a c t i o n of the swimming muscles. The s t i f f n e s s of the b e l l was measured from data i n Chapter I I . The mass was d e r i v e d from standard equations t h a t d e s c r i b e the motion of an o s c i l l a t o r . T h i s work shows that d u r i n g p e r i o d s of continuous bouts of f o r c e d c o n t r a c t i o n s , the b e l l of P o l y o r c h i s i s b e i n g f o r c e d a t i t s resonant frequency. Data i n Chapter I I I are used to h e l p v e r i f y the robustness of the model, and to estimate the e n e r g e t i c savings of working at resonance. Other advantages of u s i n g t h i s p h y s i c a l phenomenon to the mechanics of the j e t c y c l e are d i s c u s s e d . Issues r e l a t e d to the s c a l i n g of the p h y s i c a l parameters used i n the model are of obvious importance. For example, there i s p o t e n t i a l f o r s i z e dependent v a r i a t i o n of the resonant f r e q u e n c i e s of the b e l l , s i n c e as the animal grows i t s mass w i l l i n c r e a s e , and mass i s one of the major parameters c h a r a c t e r i z i n g the behaviour of an o s c i l l a t o r . These i s s u e s are not examined i n t h i s work, however, s i n c e the a nalyses made are so d e t a i l e d f o r any s i n g l e p o r t i o n of t h i s study that measurement of s i z e dependent v a r i a t i o n i n most of the data i s 8 not p u r p o s e f u l l y examined. S i z e dependent phenomenon are mentioned i n two s e c t i o n s , however. In Chapter I I , no c o r r e l a t i o n i s found between the s t r u c t u r a l s t i f f n e s s of the b e l l , and the s i z e of the b e l l . In Chapter IV, r e f e r e n c e i s made to s i z e dependent p h y s i o l o g i c a l phenomena that may be f u n c t i o n a l l y r e l a t e d to s c a l i n g of the resonant f r e q u e n c i e s of the locomotor s t r u c t u r e . B. ANATOMY OF THE BELL The anatomy of the locomotor s t r u c t u r e of P o l y o r c h i s has a l r e a d y been d e s c r i b e d i n d e t a i l ( G l a d f e l t e r , 1972). T h i s i n f o r m a t i o n i s summarized below and shown i n F i g u r e 1.1, and i s important to the d i s c u s s i o n of the mechanical p r o p e r t i e s of the locomotor s t r u c t u r e i n Chapter I I . The b e l l c a v i t y i s approximately c y l i n d r i c a l . In c r o s s - s e c t i o n the exumbrellar and subumbrellar o u t l i n e s are e s s e n t i a l l y c i r c u l a r . The subumbrellar mesoglea i s s c a l l o p e d i n t o e i g h t l o n g i t u d i n a l a d r a d i a i r i d g e s , or ' j o i n t s ' t h a t extend from above the subumbrellar summit n e a r l y to the margin of the b e l l . Most of the b e l l c o n s i s t s of t r a n s p a r e n t , n o n c e l l u l a r mesoglea that i s t r a v e r s e d by numerous, r a d i a l l y - a r r a n g e d f i b e r s . The number of f i b e r s i s g r e a t e s t near the b e l l margin and decreases g r a d u a l l y towards the apex. F i b e r d e n s i t y i s g r e a t e s t between the j o i n t s . No f i b e r s are present i n the j o i n t s . 9 C i r c u l a r muscles l i n e the subumbrellar surface and power the jet cycle. The cycle i s i n i t i a t e d by contraction of these swimming muscles, which reduces the diameter of the b e l l . Gladfelter shows that the b e l l does not change length during th i s process. Water contained i n the b e l l cavity i s ejected through the o r i f i c e surrounded by the velum, and the animal i s propelled in the opposite d i r e c t i o n . Deformation of the b e l l during t h i s process i s not uniform around the circumference, since the mesoglea folds around the adradial j o i n t s . The joint s appear to form the fulcrums of hinges, the hinges consisting of the join t s themselves and the mesoglea between them. Recoil of the b e l l to resting dimensions i s a completely passive process, powered by potential s t r a i n energy stored during the deformation of the b e l l in the e l a s t i c mesogleal tissue comprising the locomotor structure, or b e l l . 10 F i g u r e 1.1. S t r u c t u r e of the locomotor b e l l i n the hydromedusean P o l y o r c h i s (from G l a d f e l t e r , 1972). The top drawing i s a l o n g i t u d i n a l s e c t i o n through the b e l l . The o r i e n t a t i o n of the s e c t i o n i s shown i n the c r o s s - s e c t i o n beneath i t . The bottom drawing i s an expanded view of a p o r t i o n of a c r o s s - s e c t i o n . Symbols: P peduncle; RC r a d i a l c a n a l : JM j o i n t mesoglea; M manubrium; BM b e l l mesoglea; V velum; RF r a d i a l mesogleal f i b e r s ; G gonads; AR a d r a d i u s ; IR i n t e r r a d i u s ; CM c i r c u l a r muscles; RM r a d i a l muscles. 11 12 CHAPTER I I . MECHANICAL PROPERTIES OF THE LOCOMOTOR STRUCTURE A. INTRODUCTION The mesoglea of c o e l e n t e r a t e s can f u n c t i o n m e c h a n i c a l l y i n as d i v e r s e r o l e s as a h i g h l y deformable and e x t e n s i b l e body w a l l i n s e s s i l e forms such as sea anemones, or as an e l a s t i c , g e l - l i k e m a t e r i a l i n p e l a g i c j e l l y f i s h . In the l a t t e r animals, which swim by j e t p r o p u l s i o n , the mesoglea i s an i n t e g r a l p a r t of the locomotor s t r u c t u r e , and i n f o r m a t i o n on the mechanical p r o p e r t i e s e x h i b i t e d by the mesoglea i s r e q u i r e d to understand the mechanics and e n e r g e t i c s of locomotion i n t h i s animal. The mechanical and s t r u c t u r a l bases f o r the p r o p e r t i e s e x h i b i t e d by the mesoglea of sea anemones are w e l l known (Chapman, 1953a, 1953b; Alexander, 1962; G o s l i n e , 1971a, 1971b; Koehl, 1977a, 1977b), but those f o r j e l l y f i s h mesoglea have not been s t u d i e d e x t e n s i v e l y . The f i r s t and only mechanical t e s t s on j e l l y f i s h mesoglea were completed by Alexander (1964) on the scyphozoan Cyanea c a p i l l a t a . He examined v i s c o e l a s t i c p r o p e r t i e s i n i s o l a t e d samples of mesoglea, and may have had problems w i t h m a i n t a i n i n g the i n t e g r i t y of h i s p r e p a r a t i o n s , s i n c e o b s e r v a t i o n s from the present study show t h a t f l u i d s i n i s o l a t e d p i e c e s of mesoglea of the j e l l y f i s h s t u d i e d i n t h i s work e a s i l y leak from cut 13 s u r f a c e s . I t i s not p o s s i b l e to i n f e r from Alexander's work whether t h i s same problem e x i s t e d i n the i s o l a t e d mesoglea of the j e l l y f i s h he examined. Nonetheless, care was taken to a v o i d t h i s problem i n the present study. The r e s u l t s of Alexander's t e n s i l e creep experiments, however, showed that the mesoglea had a very broad d i s t r i b u t i o n of r e t a r d a t i o n times, suggesting t h a t the mesoglea resembles simple polymeric g e l s . T h i s i n f o r m a t i o n has l i t t l e f u n c t i o n a l importance i n j e l l y f i s h , s i n c e deformations of the b e l l d u r i n g locomotion are on the order of 1 sec i n d u r a t i o n , much l e s s than the broad spectrum of r e l a x a t i o n times found by Alexander. Attempts are made i n the present study to measure w i t h a n o n - d e s t r u c t i v e t e s t the dynamic mechanical p r o p e r t i e s of the whole s t r u c t u r e at r a t e s of deformation that simulate the n a t u r a l r a t e s of c o n t r a c t i o n . T h i s chapter a l s o examines, f o r the f i r s t time, the s t a t i c mechanical p r o p e r t i e s of the swimming s t r u c t u r e i n whole animals, and i n c o n j u n c t i o n w i t h other data, to i n f e r the s t a t i c p r o p e r t i e s of the mesoglea. 14 B. MATERIALS AND METHODS 1. S t a t i c Mechanical T e s t s L i v e P o l y o r c h i s p e n i c i l l a t u s were obtained from Bamfield I n l e t , on the west coast of Vancouver I s l a n d . They were maintained i n running seawater a q u a r i a u n t i l use. The animals used ranged i n s i z e from 20.5 mm to 41.5 mm ( b e l l h e i g h t ) . The s t a t i c mechanical p r o p e r t i e s of the locomotor s t r u c t u r e were examined by measuring the pressure-volume r e l a t i o n s h i p s of the subumbrellar c a v i t y . L i v e animals were a n a e s t h e t i z e d i n i s o t o n i c magnesium c h l o r i d e ( G l a d f e l t e r , 1972), and the manubrium and gonads were removed so that they would not i n t e r f e r w i t h the procedures d e s c r i b e d below. The margin of the r e l a x e d b e l l was glued (with c y a n o a c r y l a t e adhesive) to a f l a t p i e c e of P l e x i g l a s forming a leakproof s e a l at the i n t e r f a c e (see F i g u r e 2.1). The i n n e r apex of the b e l l r e s t e d on a 1 ml p l a s t i c s y r i n g e t h a t protruded through a h o l e i n the P l e x i g l a s p l a t e and i n t o the b e l l c a v i t y . T h i s prevented u n n a t u r a l l o n g i t u d i n a l s h o r t e n i n g of the b e l l . Small h o l e s d r i l l e d through the s y r i n g e b a r r e l allowed water to f l o w i n t o and out of the b e l l c a v i t y . For r e f e r e n c e i n the methods d e s c r i b e d below, a l l of t h i s w i l l be r e f e r r e d to as the support apparatus. 15 F i g u r e 2.1. Schematic drawing of the set-up used to measure the pressure-volume r e l a t i o n s h i p s of the subumbrellar c a v i t y . Symbols: C v i d e o camera; VDA vi d e o dimension a n a l y z e r ; M monitor; R ch a r t r e c o r d e r . The i n s e t shows a t y p i c a l r e c o r d made d u r i n g a s t a t i c t e s t , showing the movement of the meniscus b e f o r e and a f t e r the system was opened t o the f l u i d bath (time of opening i n d i c a t e d by the arrow). The v e r t i c a l s c a l e bar r e p r e s e n t s 1 mm of water, and the h o r i z o n t a l bar 45 seconds. 1 6 17 I t i s not p o s s i b l e to q u a n t i f y the consequences of g l u e i n g the f r e e margin of the b e l l on the pressure-volume r e l a t i o n s h i p s , because i t i s not p o s s i b l e to measure the pressure-volume r e l a t i o n s h i p s of the subumbrellar c a v i t y without c o n s t r a i n i n g the b e l l margin. T h i s c o n s t r a i n t on the movement of the b e l l margin, however, w i l l be r e l a t i v e l y small because P o l y o r c h i s i s a c y l i n d r i c a l animal, and movements of the b e l l margin at r i g h t angles to the c y l i n d r i c a l a x i s are small compared to the l e n g t h of the animal. T h i s procedure should not be used f o r s p e c i e s of j e l l y f i s h t h a t are umbrellar-shaped. The e n t i r e support apparatus was submerged i n an i s o t o n i c magnesium c h l o r i d e bath. A 3-way v a l v e was a t t a c h e d to the s y r i n g e and when opened i n t o the f l u i d bath allowed e q u i l i b r a t i o n of pressure i n t e r n a l and e x t e r n a l to the b e l l c a v i t y . The v a l v e was a l s o a t t a c h e d to e i t h e r a 5 ml or 10 ml p i p e t v i a a tygon tube (I.D. 0.5 cm). The p i p e t was i s o l a t e d and suspended v e r t i c a l l y o u t s i d e the f l u i d bath, and the e n t i r e system was f i l l e d w i t h i s o t o n i c magnesium c h l o r i d e s o l u t i o n . A small amount of detergent was added to minimize the s u r f a c e t e n s i o n of the f l u i d i n the p i p e t . A i r remaining i n the b e l l c a v i t y was removed w i t h a f i n e s y r i n g e i n s e r t e d through the t h i c k mesoglea of the penducle. The h o l e c r e a t e d i n the mesoglea d u r i n g t h i s procedure s e l f - s e a l e d . T h i s i s evident s i n c e the system c o u l d maintain a constant d e f l a t i o n pressure 18 a f t e r removal of the a i r w i t h the s y r i n g e . A c l o s e d system, c o n t a i n i n g the f l u i d i n the b e l l c a v i t y , p i p e t , and tygon tube connecting the two, was c r e a t e d by c l o s i n g the v a l v e to the f l u i d bath. The p i p e t was then lowered v e r t i c a l l y r e l a t i v e to the s u r f a c e of the water bath. The volume of f l u i d i n the p i p e t was recorded b e f o r e and a f t e r t h i s movement, and the d i f f e r e n c e between the two volumes was taken to be the volume of f l u i d removed from the b e l l c a v i t y . The removal of f l u i d from the b e l l c a v i t y d u r i n g t h i s process cre a t e d a s l i g h t p r e s s u r e d i f f e r e n c e between the f l u i d bath and the b e l l c a v i t y . A symmetrical, l a t e r a l , inward compression of the mesoglea was e n f o r c e d , which a c c u r a t e l y simulated the b e l l deformations c r e a t e d by a n a t u r a l c o n t r a c t i o n of the b e l l powered by the swimming muscles. I t should be noted t h a t t h i s procedure does not simulate the n a t u r a l r a t e s of deformation, but only the f i n a l compressions i n the mesoglea induced by the deformations. Non-symmetrical c o l l a p s e s of the b e l l o c c u r r e d o c c a s i o n a l l y when animals were not evenly mounted on the support apparatus. These t e s t s were not i n c l u d e d i n the f i n a l a n a l y s i s , s i n c e they would not simulate the f i n a l compressive f o r c e s generated i n the mesoglea d u r i n g a n a t u r a l deformation. The h e i g h t of the meniscus i n the p i p e t was recorded (see below) a f t e r the r e a d i n g had plateaued. T h i s o c c u r r e d i n about 45 to 60 seconds of e q u i l i b r a t i o n of the apparatus, and can be seen i n the i n s e t of F i g u r e 2.1. The system was then opened to 19 the f l u i d bath, and the b e l l re-expanded to i t s r e s t i n g dimensions. The meniscus re t u r n e d to i t s i n i t i a l p o s i t i o n as the pressure i n c r e a s e d to zero. The d i f f e r e n c e i n the l e v e l of the meniscus b e f o r e and a f t e r the system was opened was taken as the pressure change generated by removing the measured volume of f l u i d . T h i s process was repeated many times on each specimen examined, each time removing a d i f f e r e n t volume of f l u i d , thus g e n e r a t i n g a pressure-volume curve f o r a s i n g l e animal. Pressure-volume curves f o r e l e v e n d i f f e r e n t animals were generated u s i n g t h i s method. The changes i n the l e v e l of the meniscus were only i n the order of 1 mm of water or l e s s . Thus a v i d e o system was used to a m p l i f y i t s movements. The v i d e o camera was focused on the center of the meniscus i n the p i p e t , and arranged to g i v e a m a g n i f i c a t i o n of about 25X. Changes i n the h e i g h t of the meniscus were measured w i t h a video dimension a n a l y z e r (Model 303, Instruments f o r P h y s i o l o g y and Medicine, San Diego, CA) and recorded on a c h a r t r e c o r d e r . I estimate t h a t the system c o u l d r e l i a b l y measure movements as small as 0.05 mm, and hence -2 pr e s s u r e s as small as 0.5 Pa (1 P a s c a l (Pa) = 1 N m ). A t y p i c a l r e c o r d i s shown i n the i n s e t of F i g u r e 2.1. Upon completion of the experiment, the animal was removed from the apparatus. B e l l h e i g h t , b e l l width and b e l l t h i c k n e s s were recorded. B e l l width was measured while the animal was 20 f l a t t e n e d on a bench top. T h i s l e n g t h was taken to be equal to one h a l f of the circumference of the b e l l . 2. Dynamic Mechanical T e s t s a. I s o l a t e d Mesoglea Te s t specimens were prepared by making two p a r a l l e l l o n g i t u d i n a l s l i c e s i n the mesoglea (see F i g u r e 2.5), both from the b e l l margin to the summit of the subumbrellar c a v i t y , and a t h i r d h o r i z o n t a l s l i c e about 0.5 cm from the summit of the subumbrellar c a v i t y . One l o n g i t u d i n a l edge was then g l u e d (with c y a n o a c r y l a t e adhesive) to a 18 x 18 mm g l a s s cover s l i p and the h e i g h t , width and t h i c k n e s s of the sample were measured w i t h v e r n i e r c a l i b e r s . The other l o n g i t u d i n a l edge was then glued to a second cover s l i p , l e a v i n g only the small h o r i z o n t a l cut as unprotected s u r f a c e . T h i s sample was then p l a c e d i n a t e s t i n g apparatus ( d e s c r i b e d below) i n an o r i e n t a t i o n so t h a t dynamic en f o r c e d compressions would simulate the n a t u r a l deformations the sample would experience i n the i n t a c t b e l l , and compressed to p r e s t r a i n s of about -0.15. Mesogleal m a t e r i a l from f i v e animals was examined. Dynamic mechanical measurements were made on a f o r c e d - v i b r a t i o n t e s t i n g apparatus. The theory and o p e r a t i o n 21 of t h i s apparatus has been d e s c r i b e d elsewhere ( G o s l i n e and French, 1979, Denny and G o s l i n e , 1981 and Shadwick and G o s l i n e , 1985), but w i l l be summarized below. An e x c e p t i o n to the p r e v i o u s l y d e s c r i b e d procedure i s t h a t the e l e c t r o m a g n e t i c v i b r a t o r was d r i v e n by the n o i s e generator of a spectrum a n a l y z e r (Model 5820A Cross Channel Spectrum A n a l y z e r , Wavetek Rockland, Inc., N.J.). T h i s gave a n o i s e s i g n a l w i t h a constant power spectrum over the range of f r e q u e n c i e s s t u d i e d . The specimen was p l a c e d between a f o r c e t r a n s d u c e r , and a displacement transducer that was a t t a c h e d to the e l e c t r o m a g n e t i c v i b r a t o r . At each frequency of the f o r c e d o s c i l l a t i o n , the Spectrum A n a l y z e r computed both the r a t i o of the amplitudes of the F o u r i e r components of the f o r c e and displacement transducer s i g n a l s , and the phase s h i f t ( 6") between the two s i g n a l s . Specimen dimensions were used to c a l c u l a t e s t r e s s ( f o r c e / c r o s s - s e c t i o n a l area) from the output of the f o r c e transducer and s t r a i n (change i n l e n g t h / o r i g i n a l length) from the output of the displacement t r a n s d u c e r . The r a t i o ( s t r e s s / s t r a i n ) g i v e s the complex dynamic e l a s t i c A modulus (E ) of the t e s t specimen. E', the storage modulus, i s a measure of the energy s t o r e d e l a s t i c a l l y per c y c l e and i s c a l c u l a t e d as: A E' = E (cos S) (2.1) where £ i s the phase angle. The l o s s modulus, d e f i n e d as: 22 E'' = E ( s i n £ ) (2.2) i s a measure of the v i s c o u s energy l o s s per c y c l e . A l l c a l c u l a t i o n s were performed on a D i g i t a l Equipment C o r p o r a t i o n MINC-11/23 computer. The tangent of the phase s h i f t (tan f ) i n d i c a t e s the amount of energy l o s s r e l a t i v e to the energy s t o r e d per c y c l e (damping f a c t o r ) and can be used to c a l c u l a t e the r e s i l e n c e (R) of the m a t e r i a l w i t h the f o l l o w i n g equation (Wainwright et a l . , 1976): ln(100/R) = I V x tan (^ ) (2.3) The l e n g t h of time r e q u i r e d to c o l l e c t the dynamic data was important, s i n c e the compressed samples of mesoglea r a p i d l y l o s t t h e i r i n t e g r i t y . T h e r e f o r e , data had to be c o l l e c t e d as f a s t as p o s s i b l e . The time r e q u i r e d f o r c o l l e c t i o n of data i s dependent on the frequency span of the n o i s e generator. Data were c o l l e c t e d at the 0 to 10 Hz span, r e q u i r i n g about 10 seconds to produce a spectrum. Approximately 5 or 6 s p e c t r a were averaged to g i v e a t o t a l data c o l l e c t i o n time of about 60 seconds. The l e n g t h of time r e q u i r e d to prepare the sample and measure i t s dimensions was not so c r i t i c a l , s i n c e o b s e r v a t i o n s showed t h a t i s o l a t e d and u n d i s t u r b e d samples would remain u n a l t e r e d f o r minutes. 2 3 b. I n t a c t Locomotor S t r u c t u r e A l i v e animal was prepared as d e s c r i b e d f o r the s t a t i c mechanical t e s t s and a t t a c h e d to a s i m i l a r support apparatus. The theory and apparatus f o r the dynamic t e s t s are s i m i l a r to those d e s c r i b e d above f o r the dynamic mechanical t e s t s on the i s o l a t e d mesoglea. In the dynamic t e s t s d e s c r i b e d i n t h i s s e c t i o n , however, the d r i v e n system was f l u i d . Thus the f o r c e measurements were r e p l a c e d w i t h pressure measurements, and the displacement measurements were r e p l a c e d w i t h measurements of the changes i n volume. The volume measurements were made by c o r r e l a t i n g the displacement of the e l e c t r o m a g n e t i c v i b r a t o r w i t h volume changes i n a 50 ml ground g l a s s s y r i n g e . T h i s s y r i n g e , a t t a c h e d v i a the plunger and d r i v e n by the v i b r a t o r at one end, was att a c h e d to the bottom of the support apparatus through the p l a s t i c 1 ml s y r i n g e a t i t s other end. Thus, d y n a m i c a l l y enforced changes i n the volume of the subumbrellar c a v i t y were induced by the e l e c t r o m a g n e t i c v i b r a t o r d r i v i n g the plunger of the 50 ml ground g l a s s s y r i n g e . These volume changes were recorded by a displacement transducer t h a t f o l l o w e d the movement of the plunger of the 50 ml s y r i n g e . C a l i b r a t i o n of volume changes was performed by c o r r e l a t i n g the c a l i b r a t i o n of the displacement transducer w i t h the r e l a t i o n of v e r t i c a l 24 displacements of the plunger t o corresponding changes i n the volume of f l u i d removed from the b a r r e l of the s y r i n g e . Pressure changes i n the subumbrellar c a v i t y a s s o c i a t e d w i t h the e n f o r c e d volume changes were measured w i t h a M i l l a r M i k r o - T i p Catheter Pressure Transducer. A p i e c e of 3.6 mm (O.D.) s t a i n l e s s s t e e l t u b i n g was i n s e r t e d and glued i n t o a hole through the P l e x i g l a s p l a t e of the support apparatus. One end protruded i n t o the subumbrellar c a v i t y and was p o s i t i o n e d so that the induced movement of the b e l l would not be d i s t u r b e d . The other end was f i t t e d w i t h a p l a s t i c dome that was s u p p l i e d w i t h the tr a n s d u c e r , so that the t i p of' the transducer c o u l d be e a s i l y i n s e r t e d w h ile s e a l i n g the h o l e . The output of the pre s s u r e s i g n a l was passed through a f o u r t h order low pass a c t i v e f i l t e r w i t h i t s 3 dB c u t o f f p o i n t set a t 5 Hz. The f i l t e r e d dynamic pressure s i g n a l reached a maximum amplitude t h a t was i n the order of 1 mm of water p r e s s u r e . The e l e c t r o m a g n e t i c v i b r a t o r was not capable of d r i v i n g the 50 ml g l a s s s y r i n g e through l a r g e enough displacements to simulate a f u l l c o n t r a c t i o n of the b e l l . The use of a l a r g e r s y r i n g e would not s o l v e t h i s problem. Larger s y r i n g e s would c e r t a i n l y g i v e l a r g e r volume changes f o r the same v e r t i c a l displacement of the b a r r e l , but l a r g e r s y r i n g e s a l s o r e q u i r e l a r g e r f o r c e s to move the b a r r e l . The e l e c t r o m a g n e t i c v i b r a t o r was not capable of g e n e r a t i n g the f o r c e s r e q u i r e d t o move the 25 b a r r e l s of the l a r g e r s y r i n g e s over the displacements enforced w i t h the sm a l l e r s y r i n g e s . Thus, the b e l l was f o r c e d i n t o a s t a t i c compression to about 50% to 75% of. i t s r e s t i n g volume ( i e . near the midpoint of a normal c o n t r a c t i o n ) . T h i s was achieved by moving the el e c t r o m a g n e t i c v i b r a t o r (with a t t a c h e d plunger) r e l a t i v e to the b a r r e l of the 50 ml s y r i n g e and the att a c h e d support apparatus. The volume f l u c t u a t i o n s were then enforced a t t h i s l e v e l of the simulated s t a t i c c o n t r a c t i o n . The magnitude of these volume changes reached a maximum of about 2 mis. For comparison, the volume of the subumbrellar c a v i t y was about 8 mis. Attempts were made to enforce s m a l l e r s t a t i c compressions, but the pressures generated i n the subumbrellar c a v i t y d u r i n g the enf o r c e d dynamic volume f l u c t u a t i o n s were too low to r e c o r d . Larger enforced s t a t i c compressions c o u l d not be used, s i n c e the enforced dynamic volume f l u c t u a t i o n s caused the subumbrellar s u r f a c e to touch the p l a s t i c support s y r i n g e , thus r e s t r i c t i n g the f r e e flow of water out of the c a v i t y . T h i s caused a r a p i d , l a r g e i n c r e a s e i n the pressure t h a t d i d not r e f l e c t the r e a l dynamic pressure i n the subumbrellar c a v i t y induced by the volume f l u c t u a t i o n s . The e s t a b l i s h m e n t of the time of data c o l l e c t i o n f o r these dynamic t e s t s was not dependent on r e s t r i c t i o n s set by the leakage of f l u i d from the mesoglea, as f o r the dynamic t e s t s on the i s o l a t e d samples of mesoglea. The mesoglea was not cut i n t h i s experiment. However, i t was found to be very d i f f i c u l t to 2 6 maintain the s t a t i c compressions f o r long p e r i o d s of time, i e . i n the order of minutes. T h e r e f o r e , the data were c o l l e c t e d as f a s t as p o s s i b l e . The spectrum a n a l y z e r was set to the 5 Hz span. The time f o r data c o l l e c t i o n was about 30 seconds. V i b r a t i o n s of the support apparatus induced by the e l e c t r o m a g n e t i c v i b r a t o r i n t r o d u c e d extraneous r e s u l t s , which were accounted f o r as f o l l o w s . The magnitude of the F o u r i e r components f o r the power spectrum of the d r i v e n i s o l a t e d support apparatus were s u b t r a c t e d from the components of the spectrum f o r the data c o l l e c t e d w i t h the animal a t t a c h e d to the apparatus. The magnitudes of the F o u r i e r components of the v i b r a t i o n of the apparatus alone were at l e a s t an order of magnitude l e s s than the components w i t h the animal a t t a c h e d to the apparatus. Because t h i s magnitude c o r r e c t i o n i n v o l v e d s u b t r a c t i o n of data s e t s c a l c u l a t e d at d i f f e r e n t times, phase i n f o r m a t i o n f o r the F o u r i e r components i s not a v a i l a b l e . The r e s u l t s o b tained from t h i s experiment are analogous to the complex dynamic modulus as d e f i n e d p r e v i o u s l y f o r the dynamic t e s t s made on the i s o l a t e d p i e c e s of mesoglea. 3. Observations of Mesogleal Geometry The i n t e r n a l geometry of the mesoglea was examined d u r i n g n a t u r a l , spontaneous c o n t r a c t i o n s of the swimming s t r u c t u r e . 27 L i v e animals were r e s t r a i n e d by i n s e r t i n g t h i n metal tubes h o r i z o n t a l l y through the t i p of the t h i c k a p i c a l r e g i o n of the mesoglea. T h i s d i d not d i s t o r t the normal movement of the r e s t of the b e l l . The animal was then suspended v e r t i c a l l y i n a small g l a s s aquarium, and t h i s was p l a c e d on the stage of a Wild M5 D i s s e c t i n g Miscroscope w i t h p o l a r i z i n g o p t i c s . The specimen was r o t a t e d r e l a t i v e to the p o l a r i z e r u n t i l the d e s i r e d image was obtained. T h i s image was recorded on v i d e o tape and a frame-by-frame a n a l y s i s of s i n g l e c o n t r a c t i o n s was completed. 4. S t a t i s t i c s The s i g n i f i c a n c e of a l l the r e g r e s s i o n s was t e s t e d w i t h an a n a l y s i s of v a r i a n c e procedure, w i t h the n u l l h y p o thesis set up as H e :(3 = 0 and the a l t e r n a t e h y p o t h e s i s , : p 4 0. The H 0 may be t e s t e d by determining the F s t a t i s t i c (Zar, 1984). A l l v a l u e s of the F s t a t i s t i c shown are the c a l c u l a t e d v a l u e s w i t h t h e i r a p p r o p r i a t e degrees of freedom, and a l l are s i g n i f i c a n t to a t l e a s t P = .005. 28 C. RESULTS 1. S t a t i c T e s t s The i n d i v i d u a l p o i n t s ('*') i n F i g u r e 2.2 show t y p i c a l pressure-volume data f o r a s i n g l e animal generated by repeated measurements as d e s c r i b e d i n the M a t e r i a l s and Methods. The r e l a t i o n s h i p i s c l e a r l y n o n l i n e a r , and the data were f i t t e d to a polynomial r e g r e s s i o n , shown by the l i n e marked w i t h d o t s . S i m i l i a r polynomial r e g r e s s i o n s are shown f o r other animals. The v a r i a b i l i t y i n the i n d i v i d u a l curves i s l a r g e , and t h i s i s not s u r p r i s i n g i n l i g h t of the very small pressures that were measured. (The maximum pressures r e q u i r e d t o deform the b e l l s were a l l about 25 Pa (2.5 mm of water).) The v a r i a t i o n i n the i n d i v i d u a l data p o i n t s about the r e g r e s s i o n l i n e s (not shown) i s s i m i l i a r to the data shown f o r the l i n e marked w i t h d o t s , and a l l the r e g r e s s i o n s are s i g n i f i c a n t . The area under an i n d i v i d u a l curve i s the amount of energy r e q u i r e d to deform the t i s s u e of t h a t p a r t i c u l a r animal. T h i s area was c a l c u l a t e d by t a k i n g the d e f i n i t e i n t e g r a l of the polynomial r e g r e s s i o n s w i t h the lower l i m i t of the i n t e g r a t i o n d e f i n e d as zero and w i t h the upper l i m i t taken as the maximum volume of f l u i d removed d u r i n g an experiment. For the animal w i t h the i n d i v i d u a l data p o i n t s shown, 1.05 x 10 J of energy 29 F i g u r e 2.2. T y p i c a l pressure-volume curves f o r the subumbrellar c a v i t y generated by repeated measurements as d e s c r i b e d i n the M a t e r i a l s and Methods. The equation of the 6 l i n e marked by dots i s Y = .2645 + 2.503 x 10 x X - 3.8946 12 2 18 . 3 x 10 x X + 3.0182 x 10 x X and i s s i g n i f i c a n t (FC3,113 = 47.71). The Y i n t e r c e p t i s not s i g n i f i c a n t l y d i f f e r e n t than zero (t=.238, df = 13). The i n d i v i d u a l data p o i n t s ('*') are the pressure-volume data used t o d e r i v e t h i s r e g r e s s i o n . rd) 3 U r i S S 3 H d 31 i s s t o r e d i n the t i s s u e . T h i s a n a l y s i s was completed f o r a l l the animals and the r e s u l t s are shown i n Table 2.1. The mean -6 value f o r a l l the animals i s 4.6 x 10 J . No c o r r e l a t i o n was found between the b e l l h e i g h t and the energy storage. Pressure-volume data were converted to s t r e s s - s t r a i n data by methods d e s c r i b e d i n Appendix I. Curve B i n F i g u r e 2.3 shows the s t r e s s - s t r a i n data c a l c u l a t e d from the pressure-volume data used to generate the d o t t e d l i n e i n F i g u r e 2.2. The data were f i t t e d to polynomial r e g r e s s i o n s as d e s c r i b e d above. S t r a i n , as d e f i n e d i n Appendix I, i s the c i r c u m f e r e n t i a l s t r a i n of the e n t i r e swimming s t r u c t u r e . The s t r u c t u r a l s t i f f n e s s of the b e l l can t h e r e f o r e be d e f i n e d as the slope of t h i s s t r e s s - s t r a i n curve. Because the curve i s n o n l i n e a r , the s t i f f n e s s i s s t r a i n dependent. The s t i f f n e s s was c a l c u l a t e d by u s i n g the polynomial r e g r e s s i o n s t o p r e d i c t the s t r e s s at two p o i n t s , where the upper value of s t r a i n used to p r e d i c t the s t r e s s was taken as the maximum s t r a i n measured e x p e r i m e n t a l l y , and the lower v a l u e of the s t r a i n was taken as 0.1 s t r a i n u n i t s l e s s than t h i s maximum v a l u e . The s t i f f n e s s was then c a l c u l a t e d as a l i n e a r slope over the range d e f i n e d by the s t r a i n s . For t h i s animal, the s t r u c t u r a l s t i f f n e s s of the -2 b e l l i s 260 N m . T h i s process was repeated f o r a l l the animals, and the r e s u l t s are summarized i n Table 2.1. The mean value of the s t r u c t u r a l s t i f f n e s s f o r a l l the b e l l s i s -2 150 N m . The maximum c i r c u m f e r e n t i a l s t r a i n s (Equation A . l ) 32 F i g u r e 2.3. T y p i c a l s t r e s s - s t r a i n curves f o r the swimming s t r u c t u r e and mesoglea of hydromedusean j e l l y f i s h . The l i n e l a b e l l e d 'B' i s a polynomial r e g r e s s i o n f o r data c a l c u l a t e d by methods d e s c r i b e d i n the Appendix, where s t r a i n i s d e f i n e d as changes i n the i n s i d e r a d i u s of the subumbrellar c a v i t y , and i s the c i r c u m f e r e n t i a l s t r a i n of the swimming s t r u c t u r e . O r i g i n a l pressure-volume data were taken from F i g u r e 2.2. The l i n e l a b e l l e d 'A' has s t r a i n d e f i n e d i n terms of changes i n the i n t e r r a d i a l r e g i o n of the mesoglea and i s d e f i n e d by E quation 2.4. Both r e g r e s s i o n s were s i g n i f i c a n t and the s t a t i s t i c s f o r the l i n e s are ( s t a r t i n g from l e f t to r i g h t ) : Y = 11.06 + 442.78 x X + 2 5538.28 x X (F = 46.49; df = 2,12) and Y = 2.655 + 28.134 2 x X + 443.559 x X (F = 46.49; df = 2,12). 34 measured d u r i n g the experiments has a mean value ox -.27 (see Table 2.1). 2. Dynamic T e s t s on I s o l a t e d Mesoglea The experimental p r o t o c o l used to c o l l e c t the p r e v i o u s data i n v o l v e d an i n i t i a l r a p i d e n f o r c e d deformation of the locomotor s t r u c t u r e , f o l l o w e d by an e q u i l i b r a t i o n p e r i o d p r i o r to the a c t u a l c o l l e c t i o n of the data that l a s t e d f o r about 60 seconds. Frequency dependent mechanical p r o p e r t i e s have a l r e a d y been shown to e x i s t i n the mesoglea of i e l l y f i s h (see Alexander, 1964), thus the s t r u c t u r a l s t i f f n e s s c a l c u l a t e d u s i n g these data cannot be used d i r e c t l y to represent the dynamic s t i f f n e s s of the locomotor s t r u c t u r e , s i n c e the d u r a t i o n of the deformation of the b e l l i n a n a t u r a l c o n t r a c t i o n i s only about 1 second. For t h i s reason, dynamic t e s t s were made on i s o l a t e d p i e c e s of mesoglea at f r e q u e n c i e s that span the frequency of a n a t u r a l c o n t r a c t i o n . F i g u r e 2.4 summarizes dynamic measurements made on f i v e d i f f e r e n t i e l l y f i s h . For c l a r i t y , o n l y these three of the f i v e s ets of experiments are shown, and these show the maximum and minimum valu e s of the dynamic measurements found i n the 35 TABLE 2.1. Summary of data d e r i v e d from the pressure-volume curve and the s t r e s s - s t r a i n c urves. A = Maximum c i r c u m f e r e n t i a l s t r a i n . B = Maximum s t r a i n i n the i n t e r r a d i a l r e g i o n . F E D C A B C D E F -.26 -.36 -.37 -.32 -.37 -.31 -.15 -.18 -.14 -.34 -.23 .11 .13 . 13 .12 . 14 .12 .07 .08 .07 .13 .09 22 34 35 49 53 11 18 21 54 20 38 130 110 150 180 260 40 130 160 300 70 160 240 300 420 530 780 140 340 400 620 240 390 6.4 7.0 6.5 1.9 10.5 4.1 1.9 1.4 2.5 5.1 3.1 X -.27 -.11 32 150 400 4.6 * See the R e s u l t s f o r an e x p l a n a t i o n of a n a l y t i c a l methods. 36 experiments. Frequencies shown span the swimming frequency of the l i v e animal. Each p o i n t r e p r e s e n t s the averaged value f o r eleven measurements. R e p r e s e n t a t i v e e r r o r bars are shown, and are standard e r r o r s about the mean a t each frequency. A two way a n a l y s i s of v a r i a n c e was performed on the storage modulus data w i t h 0.2 Hz and 2.2 Hz data as treatments of one f a c t o r , and the i n d i v i d u a l s as the treatments of the other f a c t o r . No s i g n i f i c a n t d i f f e r e n c e s were found between the i n d i v i d u a l s (F = .5483, df = 4,100). From t h i s , i t was d e c i d e d to average data f o r i n d i v i d u a l s t o g i v e a s i n g l e v alue f o r the storage modulus. At a frequency of 1 Hz, the averaged storage modulus i s -2 -2 1000 N m , w i t h a standard e r r o r of 78 N m (n=63). The average damping f a c t o r f o r the same set of measurements i s 0.176, wit h a standard e r r o r of 0.006. T h i s g i v e s a r e s i l e n c e of 58%. Large d i f f e r e n c e s e x i s t between the mean value of the s t r u c t u r a l s t i f f n e s s c a l c u l a t e d from the s t a t i c t e s t s -2 (150 N m ) and the modulus c a l c u l a t e d from the dynamic t e s t s -2 (1000 N m ). These d i f f e r e n c e s are not s u r p r i s i n g , and are r e l a t e d to two problems. (1) The time i n t e r v a l i n which the measurements were made were d i f f e r e n t . (2) The deformation of the t e s t specimens themselves were d i f f e r e n t . These d i f f e r e n c e s are d e s c r i b e d next. The t e s t specimens i n the s t a t i c t e s t s were whole, i n t a c t 37 F i g u r e 2.4. Dynamic mechanical data f o r i s o l a t e d samples of j e l l y f i s h mesoglea. Each p o i n t on a l i n e r e p r e s e n t s the averaged value of ele v e n separate measurements. The r e p r e s e n t a t i v e e r r o r bars are the standard e r r o r s about the means. The s o l i d l i n e s are graphs of storage modulus and the broken l i n e s are graphs of the damping f a c t o r (tan( € ) ) . F R E Q U E N C Y | H Z ) 39 b e l l s . The s t r a i n s used i n the c a l c u l a t i o n of the s t i f f n e s s from those t e s t s were the c i r c u m f e r e n t i a l s t r a i n s of the e n t i r e locomotor s t r u c t u r e , i n c l u d i n g l a r g e deformations a s s o c i a t e d w i t h the bending of the mesoglea around the ' j o i n t s ' . Thus the s t i f f n e s s c a l c u l a t e d u s i n g that method r e p r e s e n t s the s t i f f n e s s of the e n t i r e swimming s t r u c t u r e . The t e s t specimens used i n the dynamic t e s t s , however, were p i e c e s of i s o l a t e d mesoglea, and the s t r a i n s used to c a l c u l a t e the storage moduli were a c t u a l compressive s t r a i n s i n the mesogleal m a t e r i a l . Thus, i t i s not s u r p r i s i n g that the dynamic modulus i s d i f f e r e n t than the s t a t i c s t r u c t u r a l s t i f f n e s s ; the s t a t i c t e s t s measure the p r o p e r t i e s of the s t r u c t u r e , w h ile the dynamic t e s t s measure the p r o p e r t i e s of the m a t e r i a l comprising the s t r u c t u r e . I f the deformations i n the s t a t i c t e s t s were uniform, i t would be p o s s i b l e to d i r e c t l y compare the r e s u l t s . The deformations of the b e l l d u r i n g n a t u r a l c o n t r a c t i o n s ( t h a t were simulated d u r i n g the s t a t i c t e s t s ) are not uniform. T h i s can be observed i n the deformation of the locomotor s t r u c t u r e i n l i v e animals. The importance of the d i f f e r e n c e s i n the r a t e s of the deformations of the samples i n the two experiments i s d e s c r i b e d i n the D i s c u s s i o n . 3 . Observations of Mesogleal Geometry The o u t l i n e s of two images showing the deformation of the 40 locomotor s t r u c t u r e f o r an animal i n which the records were e x c e p t i o n a l l y c l e a r are shown i n F i g u r e 2.5. The diagram l a b e l l e d 'RELAXED' i s the o u t l i n e of a r e l a x e d phase, while t h a t l a b e l l e d 'CONTRACTED' i s the o u t l i n e of a f u l l y c o n t r a c t e d animal. "1" was measured on the r e s t i n g phase, and d e f i n e s the r e s t i n g l e n g t h of a ' j o i n t ' . I t i s the d i s t a n c e from the v e r t e x of the ' j o i n t ' to the p o s i t i o n where "C" was measured. The values shown i n F i g u r e 2.5 were Used to c a l c u l a t e v a r i o u s s t r a i n s , and are d e f i n e d below. The shaded areas show the regions of the b e l l t h a t were i s o l a t e d f o r the dynamic t e s t s . Using these d a t a , c o l l e c t e d from the p o l a r i z e d l i g h t v i d e o records on spontaneously c o n t r a c t i n g animals, i t was p o s s i b l e to measure the s t r a i n i n the i n t e r r a d i a l r e g i o n s . (see F i g u r e 1.1 f o r an e x p l a n a t i o n of t h i s term). The t o t a l c i r c u m f e r e n t i a l s t r a i n was c a l c u l a t e d as i n Appendix I, where the i n s i d e r a d i u s v a l u e s are one h a l f of the v a l u e s of "A" or "A'" i n F i g u r e 2.5. The s t r a i n i n the i n t e r r a d i a l r e g i o n i s d e f i n e d by: r = ( C - C ' ) / C (2.4) ^ I where the symbols are d e f i n e d i n F i g u r e 2.5. F i g u r e 2.6 shows the r e l a t i o n s h i p between the s t r a i n i n the i n t e r r a d i a l r e g i o n r e g i o n and the t o t a l c i r c u m f e r e n t i a l s t r a i n f o r the animal i n which the o u t l i n e s of the b e l l were e x c e p t i o n a l l y c l e a r i n the 41 F i g u r e 2.5. An o u t l i n e of changes i n the i n t e r n a l geometry of the locomotor b e l l t r a c e d o f f p o l a r i z e d l i g h t v i d e o r e c o r d s . The diagram l a b e l l e d 'RELAXED' i s from a r e l a x e d phase, and t h a t l a b e l l e d 'CONTRACTED' i s from a c o n t r a c t e d phase. See the M a t e r i a l s and Methods f o r an e x p l a n a t i o n of the other l a b e l s . The shaded regions show the areas of the b e l l t h a t were removed f o r the t e s t specimens used i n the dynamic t e s t s . .0H 43 F i g u r e 2.6. The r e l a t i o n s h i p of the s t r a i n i n the i n t e r r a d i a l r e g i o n to the t o t a l c i r c u m f e r e n t i a l s t r a i n i n the swimming s t r u c t u r e . The r e g r e s s i o n of the l i n e i s Y = 0.031 + .283 x X (F = 16.08; df = 1,8). I N T E R R A D I A L C I R C U M F E R E N T I A L S T R A I N 45 v i d e o r e c o r d s . Data f o r other animals were c o l l e c t e d , but i t was found to be very d i f f i c u l t to f i n d enough c l e a r v i d e o records to generate the r e l a t i o n s h i p s s i m i l i a r to those found f o r the animal shown i n F i g u r e 2.6. Other data showed s i m i l i a r r e s u l t s to data presented here, i n that p a i r s of measurements l i e near the r e g r e s s i o n shown i n F i g u r e 2.6. The r e g r e s s i o n l i n e i n F i g u r e 2.6 was used to transform the t o t a l c i r c u m f e r e n t i a l s t r a i n s i n F i g u r e 2.3 to corresponding c i r c u m f e r e n t i a l s t r a i n s i n the i n t e r r a d i a l r e g i o n . Curve A i n F i g u r e 2.3 i s the s t r e s s versus r e c a l c u l a t e d s t r a i n i n the i n t e r r a d i a l r e g i o n c a l c u l a t e d i n t h i s manner, and a new r e g r e s s i o n was c a l c u l a t e d f o r the l i n e . T h i s a n a l y s i s was repeated f o r each animal, and a s t a t i c modulus of e l a s t i c i t y f o r the mesoglea was c a l c u l a t e d as above. The mean value f o r the s t a t i c modulus of e l a s t i c i t y of the -2 i n t e r r a d i a l mesoglea f o r a l l the animals i s 400 N m . The mean value of the dynamic storage modulus ( c a l c u l a t e d -2 p r e v i o u s l y t o be 1000 N m ) and the mean value of the s t a t i c modulus of e l a s t i c i t y of the mesoglea determined here are s t a t i s t i c a l l y d i f f e r e n t ( t = 3.45; df = 72). F i g u r e 2.7 shows the r a d i a l s t r a i n i n the i n t e r r a d i a l r e g i o n , d e f i n e d by: = ( B - B ' ) / B 46 F i g u r e 2.7. The r e l a t i o n s h i p of the r a d i a l s t r a i n i n the mesoglea to the c i r c u m f e r e n t i a l s t r a i n i n the swimming s t r u c t u r e . The r e g r e s s i o n of the l i n e i s Y = .0101 + .2802 2 x X + 3.789 x X (F = 59.9; df = 2,7). C I R C U M F E R E N T I A L S T R A I N 48 where the symbols are d e f i n e d i n F i g u r e 2.5, p l o t t e d a g a i n s t the corresponding t o t a l c i r c u m f e r e n t i a l s t r a i n . The maximum c i r c u m f e r e n t i a l s t r a i n seen i n spontaneously c o n t r a c t i n g animals was about -0.35. T h i s corresponds to a p r e d i c t e d maximum r a d i a l s t r a i n of 0.57. The r e l a t i o n s h i p between the r a d i a l s t r a i n and c i r c u m f e r e n t i a l s t r a i n was found to be n o n l i n e a r . The reason f o r t h i s n o n l i n e a r r e l a t i o n s h i p can be shown on simple g e o m e t r i c a l arguments al o n e , s i n c e f o r constant volume (of t i s s u e ) c y l i n d r i c a l systems, the r a d i a l s t r a i n has to i n c r e a s e f a s t e r than the c i r c u m f e r e n t i a l s t r a i n . The r e l a t i o n s h i p s d i s c u s s e d i n Appendix I i m p l i c i t y show these arguments. 4. Dynamic Te s t s on the I n t a c t B e l l The data d e r i v e d from the dynamic pressure-volume t e s t do not show the shape of the pressure-volume curve, but provide an estimate of the slope of the curve a t the n a t u r a l swimming frequency. No s i g n i f i c a n t v a r i a t i o n was found i n the slope over the range of f r e q u e n c i e s t e s t e d , so data were averaged to g i v e a s i n g l e v a l u e . The mean slope d e r i v e d from the dynamic 8 - 3 8 - 3 t e s t s i s 1.5 x 10 Pa m (S.E. = 0.19 x 10 Pa m , n = 14). The c o n v e r s i o n of t h i s slope i n t o a s t r u c t u r a l s t i f f n e s s t h a t would a l l o w d i r e c t comparisons to the s t a t i c s t r u c t u r a l 49 s t i f f n e s s c a l c u l a t e d above i s not a t r i v i a l problem. A r b i t r a r y i n i t i a l c o n d i t i o n s must be used i n the equations found i n Appendix I to c a l c u l a t e t r u e dynamic s t r e s s - s t r a i n v a l u e s . I t was decided, t h e r e f o r e , not to attempt to d e r i v e a dynamic s t r u c t u r a l s t i f f n e s s from these d a t a . However, the slope d e r i v e d from these t e s t s can be compared to the slope of the pressure-volume data c a l c u l a t e d from the s t a t i c t e s t s . The slope of the s t a t i c pressure-volume curves, c a l c u l a t e d from the 7 -3 polynomial r e g r e s s i o n , has a mean value of 3.2 x 10 Pa m 7 -3 ( S . E . = 0.51 x 10 Pa m , n = 11). T h i s suggests that the dynamic s t r u c t u r a l s t i f f n e s s of the i n t a c t b e l l should be about -2 f i v e times the s t a t i c s t r u c t u r a l s t i f f n e s s , or about 750 N m 50 D. DISCUSSION T h i s chapter examines, f o r the f i r s t time, the mechanical p r o p e r t i e s of the swimming s t r u c t u r e of an i n t a c t j e l l y f i s h . P r evious attempts by others f a i l e d because i t was not p o s s i b l e to deform the s t r u c t u r e i n a manner t h a t simulated n a t u r a l c o n t r a c t i o n s . The methods d e s c r i b e d here a l l o w such " n a t u r a l " deformations. As w e l l , the c i r c u m f e r e n t i a l s t r a i n s c r e a t e d i n the b e l l d u r i n g the s t a t i c t e s t s adequately span the c i r c u m f e r e n t i a l s t r a i n s induced by a n a t u r a l c o n t r a c t i o n . The s t a t i c t e s t s , however, d i d not a l l o w measurement of the dynamic mechanical behaviour of the b e l l a t the n a t u r a l swimming frequency of the animal. T h i s problem was s o l v e d i n p a r t by measuring the dynamic mechanical p r o p e r t i e s of i s o l a t e d p i e c e s of mesoglea, but the use of i s o l a t e d p i e c e s of mesoglea c r e a t e d a d d i t i o n a l problems. T h i s procedure measured the p r o p e r t i e s of the mesogleal m a t e r i a l comprising the s t r u c t u r e , d e f i n e d here as the modulus of e l a s t i c i t y , but the s t a t i c t e s t s measured the mechanical p r o p e r t i e s of the e n t i r e s t r u c t u r e , d e f i n e d as the s t a t i c s t r u c t u r a l s t i f f n e s s . The problem was r e s o l v e d w i t h a d d i t i o n a l data c o l l e c t e d from o b s e r v a t i o n s that showed the changes i n the geometry of the mesoglea i n the i n t a c t locomotor s t r u c t u r e d u r i n g a n a t u r a l c o n t r a c t i o n . T h i s allowed a c a l c u l a t i o n of the s t a t i c modulus of e l a s t i c i t y of the mesogleal m a t e r i a l from the s t a t i c s t r u c t u r a l s t i f f n e s s data. 51 T h i s c a l c u l a t i o n was then used to i n f e r the dynamic mechanical p r o p e r t i e s of the whole b e l l , and t h i s process i s d e s c r i b e d below. Attempts were a l s o made to e x p e r i m e n t a l l y measure the dynamic behaviour of the i n t a c t b e l l , and the r e s u l t s i m p l i e d t h a t the measured dynamic s t r u c t u r a l s t i f f n e s s was w i t h i n the range c a l c u l a t e d by the previous procedure. The pressure-volume data used to measure the s t o r e d s t r a i n energy were converted to s t r e s s - s t r a i n d a t a , and the s t a t i c s t r u c t u r a l s t i f f n e s s was c a l c u l a t e d f o r the b e l l , having a mean -2 value of 150 N m . On the b a s i s of geometric changes i n the i n t a c t b e l l , i t was estimated t h a t the s t a t i c modulus of e l a s t i c i t y of the i n t e r r a d i a l mesogleal m a t e r i a l i s about -2 400 N m . T h i s v a l u e i s 2.5 times s m a l l e r than the mean dynamic modulus of e l a s t i c i t y , c a l c u l a t e d to be about -2 1000 N m . I f I assume that the r e l a t i o n between the s t a t i c and dynamic moduli of the mesoglea r e f l e c t s the r e l a t i o n between the s t a t i c and dynamic s t i f f n e s s of the whole s t r u c t u r e , I would expect the dynamic s t i f f n e s s of the whole s t r u c t u r e to be at l e a s t 2.5 times l a r g e r than the s t a t i c -2 s t i f f n e s s , or about 400 N m . T h e r e f o r e the dynamic -2 s t r u c t u r a l s t i f f n e s s should l i e between 400 N m and the measured v a l u e of the dynamic modulus of the m a t e r i a l , -2 1000 N m . The attempt to e x p e r i m e n t a l l y measure the dynamic 52 s t i f f n e s s of the whole s t r u c t u r e i m p l i e s t h a t the s t a t i c and dynamic s t i f f n e s s e s are d i f f e r e n t by a f a c t o r of about 5 times, -2 imp l y i n g a dynamic s t r u c t u r a l s t i f f n e s s of about 750 N m The true dynamic s t r u c t u r a l s t i f f n e s s , t h e r e f o r e , must l i e -2 -2 between 400 N m and 1000 N m , and i t s seems l i k e l y t h a t the c o r r e c t v a l u e l i e s near the center of t h i s range. The dynamic t e s t data p r o v i d e i n f o r m a t i o n on the mechanical p r o p e r t i e s of the i s o l a t e d mesoglea. For example, the r e s i l e n c e of the mesoglea can be c a l c u l a t e d from the damping f a c t o r . The mean val u e f o r the r e s i l i e n c e was 58%, and t h i s i s l a r g e enough to suggest that the mesoglea can f u n c t i o n as an e f f e c t i v e e l a s t i c s t r u c t u r e t o antagonize the locomotor musculature. T h i s value i s probably an underestimate of the tru e r e s i l e n c e , s i n c e f l u i d s e a s i l y leak from the exposed s u r f a c e s of the i s o l a t e d mesoglea. Other i n f o r m a t i o n can a l s o be d e r i v e d from the dynamic t e s t d a t a . I t i s i n t e r e s t i n g t h a t both the storage modulus and damping f a c t o r (tan( 6 )) measured i n the dynamic t e s t s tend to i n c r e a s e as frequency i n c r e a s e s . T h i s behavior c h a r a c t e r i z e s a polymeric m a t e r i a l moving i n t o i t s t r a n s i t i o n r e g i o n , but no attempt i s made here to c h a r a c t e r i z e the molecular s t r u c t u r e u s i n g t h i s method, s i n c e i s o l a t e d samples of mesoglea do not maintain t h e i r i n t e g r i t y f o r l o n g p e r i o d s of time. These r e s u l t s agree w i t h Alexander's (1964), where he found that scyphozoan mesoglea resembles simple polymeric g e l s . 53 The magnitude of the energy required to deform the mesogleal s tructure during a contract ion was quant i f i ed using data co l l ec t ed during the simulated contract ions i n the s t a t i c t e s t s . The energy required i s the area under the pressure-volume curve, and has an average value of -6 4.6 x 10 J . Using the re la t ionsh ips defined above for the s t a t i c s t r u c t u r a l s t i f fnes s and the range of values set for the dynamic s t i f f n e s s , i t can be shown that the dynamic s t r u c t u r a l s t i f fness i s between three and seven times larger than the s t a t i c s t r u c t u r a l s t i f f n e s s . The average value of the energy required to deform the t i ssue measured i n the s t a t i c tests can be scaled up to estimate the energy required to deform the t i ssue dynamically at the natural rate of contract ion . It should be increased by the same factor as for the s t r u c t u r a l s t i f fnes ses , or by a factor of between three and seven. This -5 -5 gives a range of values between 1.4 x 10 J and 3.2 x 10 J of energy required to deform the mesoglea during a contract ion of a t y p i c a l j e l l y f i s h . These s ca l ing f a c t o r s , however, are not completely correct because they are based on estimates of the dynamic storage modulus of the m a t e r i a l . During dynamic compressions of the t i s s u e , energy w i l l be d i s s ipa ted during the deformation, and th i s energy can be estimated from the loss modulus with re la t ionsh ips defined i n Ferry (1970, page 606). At the damping l eve l s seen for the i n t e r r a d i a l mesoglea, the energy should be increased by about 28% to account for energy lo s t through d i s s i p a t i v e processes, g iv ing a range of energy 54 r e q u i r e d to deform the t i s s u e a t n a t u r a l r a t e s of c o n t r a c t i o n -5 -5 of between 1.8 x 10 J and 4.1 x 10 J . F i n a l l y , i t i s p o s s i b l e u s i n g the measured r e s i l e n c e to c a l c u l a t e the p o r t i o n of t h i s energy t h a t w i l l be a v a i l a b l e to power the r e f i l l i n g . At a r e s i l i e n c e of 58%, the energy recovered from the e l a s t i c -5 r e c o i l of a t y p i c a l mesogleal b e l l i s between 1.0 x 10 J and -5 2.4 x 10 J . I t i s i n s t r u c t i v e to ask which s t r u c t u r e s i n the b e l l a c t u a l l y s t o r e t h i s energy. In a l i v e animal, the c o n t r a c t i o n of the swimming muscles decreases the diameter of the b e l l . T h i s deformation causes an i n c r e a s e i n the t h i c k n e s s of the w a l l of the b e l l , and r a d i a l f i b e r s t h a t are embedded i n the t i s s u e are put i n t e n s i o n and a p p a r e n t l y s t o r e s t r a i n energy. These r a d i a l mesogleal f i b e r s have been i d e n t i f i e d by v a r i o u s s t a i n i n g procedures and e l e c t r o n microscopy to be ' e l a s t i c ' f i b e r s ( G l a d f e l t e r , 1972; and B o u i l l o n and Vandermeerssche, 1957). M a t e r i a l s e x h i b i t i n g t h i s type of e l a s t i c i t y are h i g h l y e x t e n s i b l e (see G o s l i n e , 1980). The l a r g e r a d i a l s t r a i n s of about 0.57 (see F i g u r e 2.7) found i n t h i s study h e l p to v e r i f y that the r a d i a l ' e l a s t i c ' f i b e r s e x h i b i t r u b b e r - l i k e e l a s t i c i t y . For comparison, collagenous f i b e r s can be extended r e v e r s i b l y to s t r a i n s of maximally on l y 0.1 (see Wainwright et a l . , 1976). In a d d i t i o n , rough c a l c u l a t i o n s as d e s c r i b e d below, show t h a t there are probably enough of the r a d i a l ' e l a s t i c ' f i b e r s present to s t o r e a l l the energy. 55 G l a d f e l t e r (1972) completed an e x t e n s i v e survey of the d i s t r i b u t i o n of the r a d i a l mesogleal f i b e r s i n P o l y o r c h i s . He -2 measured a f i b e r d e n s i t y of about 200 f i b e r s mm . Knowing th a t 1 mm t h i c k s e c t i o n s were used t o measure the f i b e r d e n s i t i e s , i t i s p o s s i b l e to c a l c u l a t e t h a t each i e l l y f i s h has about 240,000 f i b e r s i n the mesoglea. The volume of each f i b e r -15 3 i s about 3 x 10 m (with a measured diameter of 1.0 um), and -10 3 thus the t o t a l volume of the f i b e r s i s 7 x 10 m . I f a l i n e a r s t r e s s - s t r a i n curve f o r the f i b e r s i s assumed, and i f the f i b e r s have a modulus of e l a s t i c i t y s i m i l i a r to e l a s t i n 6 -2 (10 N m , see f o r example G o s l i n e , 1980), then w i t h a s t r a i n of 0.33 ( see F i g u r e 2.7, where the average c i r c u m f e r e n t i a l s t r a i n of 0.27 (Table 2.1) corresponds to a r a d i a l s t r a i n of 0.33), the energy storage per u n i t volume i n the f i b e r s would 4 -3 be 5.45 x 10 J m . Thus, the t o t a l p o t e n t i a l s t r a i n energy -5 s t o r e d i n the f i b e r s i s estimated to be 3.8 x 10 J . T h i s value can be compared to the range of value s c a l c u l a t e d above f o r the energy r e q u i r e d to deform the t i s s u e a t the n a t u r a l -5 -5 r a t e s , which was 1.8 x 10 J and 4.1 x 10 J . In l i g h t of the many assumptions made i n making these c a l c u l a t i o n s , they imply t h a t a l l , or most, of the energy s t o r e d i n the mesoglea i s capable of b e i n g s t o r e d as s t r a i n energy i n the r a d i a l mesogleal f i b e r s , and th a t i t i s l i k e l y that the r a d i a l ' e l a s t i c ' f i b e r s have a s t i f f n e s s s i m i l i a r to that of e l a s t i n . Regardless of the a c t u a l s i t e of the s t r a i n energy 56 storage, the s t a t i c t e s t s showed a n o n l i n e a r deformation f o r the i n t a c t swimming s t r u c t u r e . T h i s probably r e s u l t s from the complex deformation of the mesoglea around the ' j o i n t s ' i n the a d r a d i a l r e g i o n . The i n i t i a l low modulus r e g i o n c o u l d r e s u l t from deformations of the n o n - f i b e r e d ' j o i n t s ' , w h ile the s t i f f e r . h i g h modulus r e g i o n a t h i g h e r s t r a i n s c o u l d r e s u l t from the l o a d i n g of the mesogleal f i b e r s themselves. Regardless of the mechanism that produces the n o n l i n e a r i t y , the shape of the s t r e s s - s t r a i n curve has important i m p l i c a t i o n s f o r j e t - p r o p e l l e d swimming ( G o s l i n e and Shadwick, 1983; G o s l i n e and DeMont. 1985). These w i l l be d i s c u s s e d i n the next chapter where I examine the importance of the n o n l i n e a r s t r e s s - s t r a i n curve i n the e n e r g e t i c s of the j e t c y c l e . 57 CHAPTER I I I . ENERGETICS OF THE JET CYCLE A. INTRODUCTION Animals swim by moving v a r i o u s p a r t s of t h e i r b o dies. These f o r c e d movements do work on the environment. The mechanical energy r e q u i r e d to do t h i s work i s generated by c o n t r a c t i o n of muscles d r i v i n g the locomotor apparatus. U n f o r t u n a t e l y , i t i s d i f f i c u l t to q u a n t i f y the t o t a l mechanical cost of doing t h i s work, s i n c e some of the energy generated by the muscles does not c o n t r i b u t e d i r e c t l y to the hydrodynamic work that generates the t h r u s t , but i s used f o r other purposes, and some i s l o s t through d i s s i p a t i v e processes as heat. The energy that does not generate t h r u s t may be a s u b s t a n t i a l p o r t i o n of the t o t a l mechanical energy generated by the muscles. Thus, any d e t a i l e d study of the e n e r g e t i c s of locomotion should not be based e x c l u s i v e l y on an a n a l y s i s of g e n e r a t i o n of t h r u s t , but based on an a n a l y s i s of the t o t a l mechanical energy generated by the muscles, i n c l u d i n g both the energy used d i r e c t l y to generate the t h r u s t , and the e n e r g i e s mentioned p r e v i o u s l y . These analyses are complicated i n most animals, however, by m u s c l e - s k e l e t a l systems that generate complex temporal and s p a t i a l movements of the locomotor apparatus. These r e s t r i c t our a b i l i t i e s to make the mechanical and p h y s i o l o g i c a l measurements that are necessary f o r a 58 d e t a i l e d understanding of the e n e r g e t i c s of locomotion. These analyses are r e l a t i v e l y simple i n the hydromedusean j e l l y f i s h used i n t h i s study, s i n c e the locomotor system i s comparatively simple. The locomotor apparatus has a l r e a d y been d e s c r i b e d i n d e t a i l i n Chapter I. Most of the b e l l c o n s i s t s of t r a n s p a r e n t , n o n c e l l u l a r mesoglea that i s t r a v e r s e d by numerous r a d i a l l y arranged f i b e r s . C i r c u l a r muscles l i n e the subumbrellar s u r f a c e and power the locomotor c y c l e . The c y c l e i s i n i t i a t e d by c o n t r a c t i o n of these swimming muscles, which reduces the diameter of the b e l l . Water contained i n the b e l l c a v i t y i s e j e c t e d through the o r i f i c e surrounded by the velum and the animal i s p r o p e l l e d i n the o p p o s i t e d i r e c t i o n . R e c o i l of the b e l l to r e s t i n g dimensions i s a p a s s i v e process, powered by energy s t o r e d i n the r a d i a l f i b e r s d u r i n g the deformation of the b e l l . No muscles e x i s t t o power the re-expansion. T h i s chapter w i l l d e s c r i b e e f f o r t s to measure the mechanical e n e r g i e s generated through the e n t i r e j e t c y c l e of a hydromedusean j e l l y f i s h . The energy generated by the c o n t r a c t i o n of the subumbrellar swimming muscles performs three f u n c t i o n s : (1) i t generates the pressure i n the subumbrellar c a v i t y , (2) i t overcomes the i n e r t i a of the movement of the b e l l , and (3) i t deforms the t i s s u e . The energy a s s o c i a t e d w i t h the f i r s t component i s used d i r e c t l y t o generate the t h r u s t t h a t w i l l p r o p e l the animal. The second and t h i r d 59 components do not c o n t r i b u t e d i r e c t l y to the g e n e r a t i o n of the t h r u s t , but are nonetheless e s s e n t i a l f o r the c y c l e t o proceed. The i n e r t i a l component makes both n e g a t i v e and p o s i t i v e c o n t r i b u t i o n s t o the g e n e r a t i o n of the j e t , s i n c e d u r i n g the i n i t i a l a c c e l e r a t i o n of the b e l l inwards, the i n e r t i a tends to a c t a g a i n s t the c o n t r a c t i o n of the swimming muscles, but when the muscles are r e l a x i n g near the end of the c o n t r a c t i o n phase, the i n e r t i a a c t s w i t h the muscles. The s t o r e d e l a s t i c energy w i l l be used to power the r e f i l l i n g of the b e l l when the swimming muscles r e l a x , and t h e r e f o r e a l l the energy r e q u i r e d to power the j e t c y c l e i s generated d u r i n g the c o n t r a c t i o n phase. 60 B. MATERIALS AND METHODS The j e l l y f i s h used i n t h i s study, the hydromedusean P o l y o r c h i s p e n i c i l l a t u s were ob t a i n e d from Bamfield I n l e t , on the west coast of Vancouver I s l a n d . They were maintained i n running seawater a q u a r i a u n t i l use. The purpose of the experimental p o r t i o n of t h i s study was to o b t a i n a c c u r a t e records of the pressure and volume changes o c c u r r i n g i n the subumbrellar c a v i t y d u r i n g spontaneous c o n t r a c t i o n s of the swimming muscles. The procedures d e s c r i b e d below have been used i n a s i m i l i a r a n a l y s i s of the e n e r g e t i c s of the j e t c y c l e i n s q u i d ( G o s l i n e and Shadwick, 1983). L i v e P o l y o r c h i s were a n a e s t h e t i z e d i n i s o t o n i c magnesium c h l o r i d e ( G l a d f e l t e r , 1972) and the manubrium and gonads were removed. These animals were then t e t h e r e d by g l u e i n g the apex of the b e l l (with c y a n o a c r y l a t e adhesive) t o a s o l i d base. Pressure records were obtained by i n s e r t i n g a seawater f i l l e d p o l y e t h y l e n e c a t h e t e r tube (PE 190) through the velum ap e r t u r e and i n t o the subumbrellar c a v i t y . The end of the tube was p l a c e d as c l o s e to the base of the peduncle as p o s s i b l e , without coming i n t o p h y s i c a l contact w i t h i t . The PE tube was a t t a c h e d at the other end to a Narco T e l e c a r e RP-1500i pressure t r a n s d u c e r . The transducer was c a l i b r a t e d a g a i n s t known s t a t i c p r e s s u r e s , and the dynamic response was determined by 61 m o n i t o r i n g f r e e resonant v i b r a t i o n s from a pressure t r a n s i e n t a p p l i e d at the c a t h e t e r t i p (Gabe, 1972). The resonant frequency was 15 Hz, w i t h a damping f a c t o r of 0.162. The s i g n a l from the transducer was c o n d i t i o n e d w i t h a Gould (Model 13-4615-50) Tranducer A m p l i f i e r and a m p l i f i e d w i t h a Gould Medium Gain D.C. P r e a m p l i f i e r (Model 13-4615-10). The s i g n a l was f i l t e r e d w i t h an o n - l i n e 15 Hz f i l t e r i n s t a l l e d i n the p r e a m p l i f i e r , which may i n t r o d u c e e r r o r s i n t o dynamic s i g n a l s . In order to account f o r these e r r o r s , the c h a r a c t e r i s t i c s of the f i l t e r were measured w i t h a Wavetek Model 5820A Cross Channel Spectrum A n a l y z e r ( R o c k l e i g h , New J e r s e y ) . Procedures f o r the c o r r e c t i o n s are d e s c r i b e d below. The f i l t e r e d s i g n a l was recorded on a Hewlett-Packard i n s t r u m e n t a t i o n tape r e c o r d e r (Model 3964A). Measurements of the subumbrellar c a v i t y volume were obtained by m o n i t o r i n g the i n t e r n a l diameter w i t h a v i d e o measuring system. T h i s system uses a Video Dimension A n a l y z e r (VDA) (Model 303, Instruments f o r P h y s i o l o g y and Medicine, San Diego, C a l i f o r n i a ) t h a t p r o v i d e s an e l e c t r i c a l s i g n a l which i s p r o p o r t i o n a l t o the s e p a r a t i o n of two c o n t r a s t boundaries on any h o r i z o n t a l l i n e i n the v i d e o image. The b e l l of the j e l l y f i s h i s t r a n s p a r e n t ; thus i t was p o s s i b l e t o a d j u s t the i n t e n s i t y of the i n c i d e n t l i g h t i n such a way t h a t the VDA t r i g g e r e d o f f the i n s i d e s u r f a c e of the locomotor b e l l . Since the subumbrellar c a v i t y i s e s s e n t i a l l y c y l i n d r i c a l i n shape. 62 w i t h a c i r c u l a r c r o s s - s e c t i o n ( G l a d f e l t e r , 1972), diameter measurements were e a s i l y converted i n t o volume measurements. The s i g n a l of the VDA passes through a 15 Hz f i l t e r and, t h i s may i n t r o d u c e e r r o r s i n dynamic s i g n a l s . In order to account f o r these e r r o r s , the c h a r a c t e r i s t i c s of the f i l t e r were measured w i t h the Spectrum A n a l y z e r . Procedures f o r the c o r r e c t i o n s are d e s c r i b e d below. The output of the VDA was recorded s i m u l t a n e o u s l y w i t h the pressure records as d e s c r i b e d above. S e l e c t e d segments of the analogue p r e s s u r e and VDA records were d i g i t i z e d on a D i g i t a l Equipment C o r p o r a t i o n MINC-11/23 computer and analyzed as d e s c r i b e d below. The pressures measured i n the subumbrellar c a v i t y were v e r y s m a l l , thus the s i g n a l - t o - n o i s e r a t i o was improved by d i g i t a l s i g n a l a v e r a g i n g . A p a r t i c u l a r l y c l e a n sequence of pressure and VDA records from 13 c o n t r a c t i o n s of the swimming muscles were d i g i t i z e d at a -1 r a t e of 200 p o i n t s sec and s i g n a l averaged. I t was found t h a t the f i n a l s i g n a l averaged pressure and VDA waveforms were s l i g h t l y asynchronous, i n that the p o s i t i o n i n time where the pressure goes n e g a t i v e was s h i f t e d by about 50 msec from where the b e l l s t a r t e d to open at the b e g i n n i n g of the r e f i l l i n g phase. These events should occur s i m u l t a n e o u s l y . E r r o r s i n t r o d u c e d by the f i l t e r s i n both the p r e s s u r e a m p l i f i e r and the VDA caused s m a l l temporal s h i f t s i n the two s i g n a l s , and t h i s r e s u l t e d i n the s l i g h t asynchrony i n the two waveforms. 6 3 To c o r r e c t f o r t h i s problem, both the s i g n a l averaged pressure and VDA records were passed through a F o u r i e r a n a l y s i s . The c h a r a c t e r i s t i c s of the f i l t e r s i n s t a l l e d i n the Gould p r e - a m p l i f i e r and the VDA were measured, and r e l e v a n t parameters were programmed to a p p r o p r i a t e l y c o r r e c t each of the harmonics i n the power spectrum of both the p r e s s u r e and VDA s i g n a l s . In a d d i t i o n , the dynamic responses of the pressure transducer and the PE t u b i n g a t t a c h e d to i t were measured and programmed to make an a d d i t i o n a l c o r r e c t i o n t o each harmonic of the pressure waveform. T h i s procedure i s d e s c r i b e d i n McDonald (1974). These c o r r e c t e d power s p e c t r a were then r e - s y n t h e i z e d i n t o t h e i r o r i g i n a l time domains, and these c o r r e c t e d waveforms were used f o r f u r t h e r a n a l y s i s , as e x p l a i n e d i n the R e s u l t s . 64 C. RESULTS 1. The C o n t r a c t i o n Phase A l l of the energy r e q u i r e d to power the j e t c y c l e i s generated by the c o n t r a c t i o n of the swimming muscles d u r i n g t h i s phase. T h i s energy can be q u a n t i f i e d by measuring s e p a r a t e l y the e n e r g i e s a s s o c i a t e d w i t h the three components l i s t e d i n the I n t r o d u c t i o n . The e n e r g i e s of the f i r s t two components can be measured from records of the p r e s s u r e and diameter changes of the subumbrellar c a v i t y t h a t occur d u r i n g a c o n t r a c t i o n of the muscles. These measurements w i l l be d e s c r i b e d next. The e n e r g i e s a s s o c i a t e d w i t h the deformation of the b e l l were measured i n the p r e v i o u s chapter and w i l l be d e s c r i b e d l a s t . F i g u r e 3.1 shows records of the o r i g i n a l pressure and diameter changes that were recorded w i t h the procedure d e s c r i b e d i n the M a t e r i a l s and Methods. These data were processed as d e s c r i b e d above, and a s i g n a l averaged, F o u r i e r c o r r e c t e d pressure-volume loop as shown i n F i g u r e 3.2 was generated. The b e l l h e i g h t of the animal used i n t h i s p o r t i o n of the study was 3.0 cm, and i t s mass was 7.3 g. The pressure-volume r e c o r d shows the events that occur i n 65 F i g u r e 3.1. T y p i c a l t r a c i n g s of p r e s s u r e and diameter data used to generate the pressure-volume curve of F i g u r e 3.2. 2 ° 5 x« to u-o o X 3> 67 F i g u r e 3 . 2 . A p r e s s u r e - v o l u m e l o o p f o r the l o c o m o t o r c y c l e of a hydromedusean j e l l y f i s h . The s o l i d l i n e r e p r e s e n t s a t r a i n of 13 p u l s e s t a k e n randomly from a l o n g t r a i n o f c o n t i n u o u s c o n t r a c t i o n s and p r o c e s s e d as d e s c r i b e d i n the M a t e r i a l s and M e t h o d s . The arrows i n d i c a t e t h e d i r e c t i o n of the c o n t r a c t i o n . 69 the middle of a long t r a i n of continuous c o n t r a c t i o n s . The swimming muscles s t a r t to c o n t r a c t near the p o i n t marked w i t h an a s t e r i s k , and the c o n t r a c t i o n phase of the c y c l e i s i n i t i a t e d . As the c o n t r a c t i o n proceeds, the pressure i n the c a v i t y s t a r t s to r i s e and peaks a t about 43 Pa. T h i s occurs when about 65% of the t o t a l volume of e j e c t e d water has been removed. The p r e s s u r e then drops o f f to zero and the c o n t r a c t i o n phase of the c y c l e i s complete. The area under the curve r e p r e s e n t s the t o t a l amount of energy that the swimming muscles produce i n g e n e r a t i n g the pressure i n the subumbrellar c a v i t y and i s 5.4 x 10 J (see Table 3.1). I n e r t i a l f o r c e s a s s o c i a t e d w i t h the a c c e l e r a t i o n s of the body w a l l cannot be measured d i r e c t l y from F i g u r e 3.2. Estimates of the a c c e l e r a t i o n of the body w a l l , however, which are necessary f o r the c a l c u l a t i o n of the i n e r t i a l f o r c e , can be d e r i v e d from the o r i g i n a l diameter data used to c a l c u l a t e the volume data shown i n F i g u r e 3.2. In order to be c o n s i s t e n t w i t h other data used i n the f o l l o w i n g c a l c u l a t i o n of the i n e r t i a l f o r c e , i e . the e f f e c t i v e mass, the diameter data has been converted to circumference d a t a , and t h e r e f o r e displacement i s expressed as a change i n the circumference of the b e l l w a l l . The second d e r i v a t i v e of these circumference measurements i s a d i r e c t measure of the a c c e l e r a t i o n of the body w a l l . T h i s can be c a l c u l a t e d by f i r s t measuring the instantaneous slope between two s u c c e s s i v e d i g i t i z e d p o i n t s (5 70 TABLE 3.1 Component of T o t a l Energy-Generated by the Muscles Energy (J) -5 (x 10 ) Percent of T o t a l CONTRACTION PHASE Pressure i n subumbrellar c a v i t y I n e r t i a of w a l l (minimum) (maximum) Deformation of the w a l l (minimum) (maximum) TOTAL (mimimum) (maximum) 5.4 7 5 8 1 8.9 14.0 61 (min.) 39 (max.) 19 32 20 29 REFILLING PHASE Pressure i n subumbrellar c a v i t y I n e r t i a of w a l l (minimum) (maximum) 1.3 3.5 8.2 76 (min.) 62 (max.) 21 39 TOTAL (minimum) (maximum) 1.7 2.1 71 msec a p a r t ) . T h i s i s the f i r s t d e r i v a t i v e of the circumference curve. The process i s then repeated on the f i r s t d e r i v a t i v e d ata, and t h i s p r o v i d e s the second d e r i v a t i v e , or the a c c e l e r a t i o n of the body w a l l . Small f l u c t u a t i o n s i n the i n i t i a l diameter curve w i l l be a m p l i f i e d u s i n g t h i s method, t h e r e f o r e , a f i v e step running average was Used on the f i r s t and second d e r i v a t i v e data to smooth out any of these small p e r t u r b a t i o n s . The mass of the body t h a t i s being a c c e l e r a t e d a l s o has to be known to c a l c u l a t e the i n e r t i a l f o r c e , s i n c e the i n e r t i a l f o r c e i s equal to the product of the a c t u a l mass and the a c c e l e r a t i o n . The mass of the a c c e l e r a t e d body i n t h i s case i n c l u d e s the mass of the a c c e l e r a t e d p o r t i o n of the animal and the mass of both the water contained i n the subumbrellar c a v i t y and the water t h a t surrounds the b e l l t h a t i s a c c e l e r a t e d d u r i n g the c o n t r a c t i o n of the b e l l . T h i s e f f e c t i v e mass cannot be measured d i r e c t l y , but can be p r e d i c t e d u s i n g data from Chapter 4 where the locomotor s t r u c t u r e of the j e l l y f i s h i s modelled as a h a r m o n i c a l l y f o r c e d o s c i l l a t o r . The model p r e d i c t s that the e f f e c t i v e mass should be between 2.5 and 6.2 times l a r g e r than the mass of the animal. Using these v a l u e s , the e f f e c t i v e mass i n t h i s study i s p r e d i c t e d t o be between 0.0183 and 0.0453 kg. The i n e r t i a l f o r c e can be c a l c u l a t e d by m u l t i p l y i n g a t each d i g i t i z e d p o i n t i n time an e f f e c t i v e mass and the a c c e l e r a t i o n of the body w a l l . 7 2 The i n e r t i a l f o r c e s c a l c u l a t e d u s i n g t h i s method are not the true i n e r t i a l f o r c e s t h a t the w a l l s experience d u r i n g the j e t c y c l e . T h i s i s because the r e a l e f f e c t i v e mass of the o s c i l l a t i n g b e l l must f l u c t u a t e over the d u r a t i o n of the c o n t r a c t i o n , s i n c e the volume of f l u i d i n the subumbrellar c a v i t y changes d u r i n g the j e t c y c l e . The e f f e c t i v e mass i s thus an average v a l u e , and the temporal changes of the t r u e i n e r t i a l f o r c e s cannot be measured. The v a l u e s c a l c u l a t e d provide an averaged estimate of the i n e r t i a l f o r c e s , and these w i l l be used i n comparisons w i t h the t o t a l e n e r g i e s generated i n the other components l i s t e d above. The purpose of t h i s a n a l y s i s i s to compare the components of the e n e r g i e s generated by the swimming muscles. Thus the i n e r t i a l f o r c e c a l c u l a t e d as the product of the e f f e c t i v e mass and the a c c e l e r a t i o n was converted to i n e r t i a l 'pressure' by d i v i d i n g the i n e r t i a l f o r c e by the instantaneous s u r f a c e area of the i n s i d e of the subumbrellar c a v i t y , s i n c e t h i s i s the area that the c a l c u l a t e d f o r c e was working on. F i g u r e 3.3 shows the c a l c u l a t e d i n e r t i a l 'pressures' generated d u r i n g the c o n t r a c t i o n and r e f i l l i n g c y c l e , w i t h the lower range of the e f f e c t i v e mass used i n the c a l c u l a t i o n of the i n e r t i a l f o r c e . The 'pressures' change s i g n a t v a r i o u s stages i n the c y c l e . These i n d i c a t e t h a t the s i g n of the a c c e l e r a t i o n of the body w a l l i s changing. The e n e r g i e s a s s o c i a t e d w i t h the i n e r t i a of the w a l l can be measured d i r e c t l y o f f these curves as the area 73 F i g u r e 3.3. I n e r t i a l 'pressures' generated d u r i n g the c o n t r a c t i o n of the locomotor muscles. The mass of the a c c e l e r a t e d body was taken to be 0.0183 kg. The arrows i n d i c a t e the d i r e c t i o n of the c o n t r a c t i o n . 74 75 under the lo o p s . The net energy used was taken as the sum of the p o s i t i v e and the negative c o n t r i b u t i o n s of the areas under the curve. The net energy a s s o c i a t e d w i t h the i n e r t i a of the -6 w a l l d u r i n g the c o n t r a c t i o n i s 1.7 x 10 J w i t h the lower range f o r the e f f e c t i v e mass used i n the c a l c u l a t i o n of the -6 i n e r t i a l f o r c e , and 4.5 x 10 J w i t h the upper range of the e f f e c t i v e mass used i n the c a l c u l a t i o n . These c a l c u l a t i o n s are the f i r s t r ecords of the i n e r t i a of the body w a l l a s s o c i a t e d w i t h i t s movement d u r i n g the j e t c y c l e of any j e t - p r o p e l l e d animal. A p o r t i o n of the energy generated by the c o n t r a c t i o n of the muscles i s used to deform the t i s s u e . Data on the q u a n t i t i t e s of energy r e q u i r e d t o deform the t i s s u e were c o l l e c t e d i n the pre v i o u s chapter. The q u a n t i t y of energy i s dependent on the r a t e i n which the t i s s u e i s deformed, and i t was d i f f i c u l t to measure the energy r e q u i r e d to deform the t i s s u e at the n a t u r a l r a t e of c o n t r a c t i o n . The q u a n t i t y of energy r e q u i r e d , however, was estimated to be between -5 -5 1.8 x 10 and 4.1 x 10 J . 2. The R e f i l l i n g Phase The c o n t r a c t i o n phase ends as the pressure i n the subumbrellar c a v i t y goes n e g a t i v e and the b e l l s t a r t s to 76 re-expand. The energy to power the r e f i l l i n g of the subumbrellar c a v i t y comes e x c l u s i v e l y from the p o t e n t i a l energy s t o r e d as s t r a i n energy i n the deformed b e l l . The t o t a l energy used can be measured u s i n g s i m i l i a r methods as those d e s c r i b e d above f o r the c o n t r a c t i o n . The energy used to generate the pressure i n the subumbrellar c a v i t y d u r i n g r e f i l l i n g i s equal to the area under the r e f i l l i n g phase of the loop i n F i g u r e 3.2. I n e r t i a l 'pressures' were c a l c u l a t e d as b e f o r e , and these data are shown i n F i g u r e 3.3. The energy used i n g e n e r a t i n g -5 the pressure i n the subumbrellar c a v i t y i s 1.3 x 10 J , while the net energy used i n g e n e r a t i n g the i n e r t i a l 'pressure' i s -6 -6 between 3.5 x 10 and 8.2 x 10 J . The t o t a l energy used i n r e f i l l i n g the subumbrellar c a v i t y can be approximated by simply adding the e n e r g i e s a s s o c i a t e d w i t h the i n e r t i a l 'pressure' and the subumbrellar c a v i t y p r e s s u r e . T h i s energy i s between -5 -5 1.7 x 10 and 2.1 x 10 J . These r e s u l t s are summarized i n Table 3.1. 77 D. DISCUSSION T h i s chapter d e s c r i b e s measurements of the mechanical energies generated d u r i n g the j e t c y c l e . The cost of t h i s locomotion cannot be q u a n t i f i e d by simply measuring the cost of g e n e r a t i n g t h r u s t . C e r t a i n l y a p o r t i o n of the mechanical energy generated d u r i n g the c o n t r a c t i o n of the swimming muscles does work on the environment and generates the t h r u s t , but some of the mechanical energy performs other f u n c t i o n s that do not do work on the environment, but are nonetheless e s s e n t i a l f o r the c y c l e t o proceed. A l l of these events should be taken i n t o account i n an a n a l y s i s of the e n e r g e t i c s of locomotion of t h i s animal. The c a l c u l a t i o n of a l l the mechanical e n e r g i e s generated by the c o n t r a c t i o n of the swimming muscles i s r e l a t i v e l y easy i n the hydromedusean j e l l y f i s h s t u d i e d here. T h i s i s because the geometry of the locomotor system of t h i s animal i s comparatively simple. The mechanical e n e r g i e s generated by the muscles have been p a r t i t i o n e d by methods d e s c r i b e d i n the R e s u l t s , and the data are summarized i n Table 3.1. In the c o n t r a c t i o n phase of the j e t c y c l e , the l a r g e s t component of the t o t a l energy i s a s s o c i a t e d w i t h the pressure-volume changes -5 i n the subumbrellar c a v i t y . T h i s process r e q u i r e s 5.4 x 10 J of mechanical energy, and r e p r e s e n t s between 39% and 61% of the 78 t o t a l mechanical energy generated d u r i n g the c o n t r a c t i o n . The energy a s s o c i a t e d w i t h the i n e r t i a of the b e l l , between -5 -5 1.7 x 10 and 4.5 x 10 J , i s c o n s i d e r a b l y s m a l l e r . I t accounts f o r o n l y between 19% and 32% of the t o t a l energy generated. The amount of energy r e q u i r e d to deform the -5 -5 locomotor s t r u c t u r e i s between 1.8 x 10 J and 4.1 x 10 J , and t h i s r e p r e s e n t s between 20% and 29% of the t o t a l mechanical energy generated d u r i n g the c o n t r a c t i o n . A simple summation of these three components g i v e s an approximate value of the t o t a l mechanical enercry t h a t i s used to generate the j e t , and t h i s -5 -4 summation y i e l d s a value between 8.9 x 10 J and 1.4 x 10 J . The importance of the r e l a t i v e magnitudes of the three components w i l l be d i s c u s s e d i n the next chapter. These v a l u e s can now be compared to previous independent estimates of power requirements f o r j e t p r o p u l s i o n i n f o u r other s p e c i e s of hydromedusae, Gonionemus ve r t e n s and Stomotoca  a t r a ( D a n i e l , 1985) and Chelophyes and A b y l o p s i s (Bone and Trueman, 1982). D a n i e l measured the oxygen consumption of swimming medusea as a f u n c t i o n of swimming frequency and was able to measure the power requirements of t h i s p r o c e s s . He a l s o p r e d i c t e d the power requirements w i t h a model based on the balance of f o r c e s the animals must experience d u r i n g the locomotion, assuming a muscle e f f i c i e n c y of between 0.1 and 0.2. His p r e d i c t e d and measured v a l u e s of the power requirements were i n good agreement. The v a l u e s f o r energy 79 from Table 3.1 can be used t o measure the e q u i v a l e n t power requirements. The e n t i r e c y c l e was completed i n about 0.8 -5 seconds, g i v i n g a power output of between 8.8 x 10 W and -4 1.4 x 10 W. To c o r r e c t f o r s i z e d i f f e r e n c e s , D a n i e l d i v i d e d the power requirements by mass to the power of 5/3. For measured power requirements from Table 3.1, t h i s g i v e s a value -5/3 of between 0.33 and 0.52 W kg . T h i s v a l u e f a l l s n i c e l y i n the c l u s t e r of data of h i s F i g u r e 6, where the range of valu e s -5/3 i s between about 0.2 and 0.75 W kg Bone and Trueman (1982) measured the t o t a l work per c y c l e as simply the pressure i n the subumbrellar c a v i t y times the volume of water e l e c t e d . They measured the pressure i n the subumbrellar c a v i t y u s i n g s i m i l i a r methods as d e s c r i b e d f o r t h i s work. Volume changes were measured by i n t e g r a t i o n of the pressure p u l s e . They argued t h a t t h i s method i s p o s s i b l e s i n c e j e t e f f l u x v e l o c i t y d u r i n g the exhalent phase depends upon the d i f f e r e n c e between chamber pressure and ambient p r e s s u r e . Volume changes w i l l n e c e s s a r i l y f o l l o w p r e s s u r e changes, and can be obtained i n d i r e c t l y by i n t e g r a t i o n of the pressure p u l s e . T h e i r estimated power outputs u s i n g t h i s method were -5 -4 c a l c u l a t e d t o be between 3.24 x 10 and 1.35 x 10 W f o r -4 -4 A b y l o p s i s and between 3.88 x 10 and 7.76 x 10 W f o r Chelophyes. These estimates agree reasonably w e l l w i t h -5 estimates of the power output f o r P o l y o r c h i s (8.8 x 10 and -4 1.4 x 10 W). They d i d not pr o v i d e the masses of the animals 80 used i n t h e i r study, so comparisons s i m i l i a r t o those made w i t h D a n i e l ' s work cannot be made. How do the s t r e s s e s generated by the swimming muscles d u r i n g the c y c l e compare to s t r e s s e s generated by other muscles? The maximum s t r e s s generated by the swimming muscles can be approximated by u s i n g the equation f o r the c i r c u m f e r e n t i a l s t r e s s i n a t h i n w a l l e d c y l i n d e r , <S~ = (P x r ) / t , where P i s the pressure i n the c y l i n d e r , r i s the r a d i u s of the c y l i n d e r , and t i s the t h i c k n e s s of the w a l l of the c y l i n d e r . T h i s c a l c u l a t i o n w i l l n e c e s s a r i l y be an underestimate of the r e a l s t r e s s generated by the swimming muscles, s i n c e t h i s c a l c u l a t i o n assumes that the swimming muscles only do work to generate the pressure i n the c a v i t y , and i t has a l r e a d y been shown t h a t t h i s i s not t r u e . The maximum pre s s u r e generated d u r i n g the c o n t r a c t i o n i s about 40 Pa (see F i g u r e 3.2), and t h i s i s reached a t a r a d i u s of about -3 6.3 x 10 m. The t h i c k n e s s of the subumbrellar l a y e r of swimmincr muscles was taken from G l a d f e l t e r (1972) to be -6 2 x 10 m. S u b s t i t u t i o n of these v a l u e s i n t o the equation f o r the c i r c u m f e r e n t i a l s t r e s s i n a t h i n w a l l e d c y l i n d e r y i e l d a 5 -2 maximum s t r e s s of about 1.25 x 10 N m . Using s i m i l i a r methods. Bone and Trueman (1982) p r e d i c t e d a maximum s t r e s s of 5 -2 about 2 x 10 N m f o r Chelophyes. For comparison, the maximum i s o m e t r i c s t r e s s of almost a l l s t r i a t e d muscles f a l l s 5 5 - 2 i n the range 3 x 10 to 5 x 10 N m (Alexander and Goldspink, 81 1977) . A s u b s t a n t i a l p r o p o r t i o n of the t o t a l mechanical energy generated i n t h i s process i s used to deform the t i s s u e d u r i n g the c o n t r a c t i o n . T h i s energy might a t f i r s t appear t o have been used a t the expense of g e n e r a t i n g mechanical energy that could have done u s e f u l hydrodynamic work. T h i s i s not t r u e , however, and the reasons w i l l be c l e a r when the mechanics of such an e l a s t i c storage system are e x p l a i n e d below. The mechanical s i g n i f i c a n c e of e l a s t i c s t r a i n energy storage systems i n j e t - p r o p e l l e d animals i s noteworthy, and has a l r e a d y been d e s c r i b e d i n some d e t a i l f o r j e t - p r o p e l l e d swimming i n squ i d ( G o s l i n e and Shadwick, 1983; G o s l i n e and DeMont, 1985). The e l a s t i c energy storage system i n squ i d mantle play s a very important r o l e i n the swimming mechanics. The locomotor system i s me c h a n i c a l l y s i m i l i a r to the j e l l y f i s h s t u d i e d i n t h i s work. F i r s t l y , i t i s g e o m e t r i c a l l y s i m i l i a r , i n t h a t f l u i d c o n t a i n e d i n a t h i c k w a l l e d c y l i n d r i c a l chamber i s e x p e l l e d through an o r i f i c e a t one end, g e n e r a t i n g a j e t tha t p r o p e l s the animal i n the opp o s i t e d i r e c t i o n . The e j e c t i o n of the water i s caused by the c o n t r a c t i o n of c i r c u l a r muscles i n the w a l l of the chamber, which decreases the diameter of the w a l l . There i s a concomitant i n c r e a s e i n the w a l l t h i c k n e s s . T h i s occurs because the squ i d mantle, l i k e the b e l l of the j e l l y f i s h , i s a constant volume system, and there 82 i s a n e g l i g i b l e change i n the l e n g t h of the c y l i n d e r d u r i n g the c o n t r a c t i o n . Secondly, d u r i n g the c o n t r a c t i o n of the c i r c u l a r muscles i n the squ i d mantle, e l a s t i c r a d i a l f i b e r s embedded i n the mantle w a l l are put i n t e n s i o n as the w a l l t h i c k n e s s i n c r e a s e s , and s t r a i n energy i s s t o r e d . These e l a s t i c s t r u c t u r e s s t o r e the s t r a i n energy a t a time i n the j e t c y c l e when the f u l l mechanical output of the muscles cannot be used to generate hydrodynamic t h r u s t . T h i s i s t r u e on g e o m e t r i c a l arguments al o n e , s i n c e by v i r t u e of the c y l i n d r i c a l shape of the locomotor system, the volume of f l u i d e x p e l l e d d u r i n g the c o n t r a c t i o n phase of the j e t c y c l e decreases as the square of the r a d i u s , and the a b i l i t y of the muscle to impart energy to the c o n f i n e d f l u i d decreases as the ra d i u s decreases. T h e r e f o r e , the p o t e n t i a l f o r doing hydrodynamic work i n a c y l i n d r i c a l system must decrease, even though the p o t e n t i a l f o r muscles to do work remains unchanged. Thus, s i n c e s q u i d probably keep the muscles a c t i v e d u r i n g the e n t i r e c o n t r a c t i o n phase, they are a b l e to i n c r e a s e the t o t a l output of t h e i r muscles by s t o r i n g the e x t r a output i n the e l a s t i c s t r u c t u r e s of the mantle. In s q u i d , i t has been proposed t h a t a n o n l i n e a r s t r e s s - s t r a i n curve f o r the e l a s t i c s t r u c t u r e i n the mantle w a l l would be the most f u n c t i o n a l of des i g n s , as the i n i t i t a l low modulus r e g i o n would a l l o w the mantle to deform e a s i l y when the p o t e n t i a l hydrodynamic output i s h i g h , but near the end of the c o n t r a c t i o n the i n c r e a s i n g 83 s t i f f n e s s of the e l a s t i c s t r u c t u r e would a l l o w the storage of s t r a i n energy when the p o t e n t i a l f o r hydrodynamic work i s low. T h i s p o t e n t i a l energy s t o r e d i n the e l a s t i c s t r u c t u r e would then he used l a t e r i n the c y c l e to power the re-expansion of the mantle, without s i g n i f i c a n t l o s s of hydrodynamic t h r u s t . In s q u i d , d u r i n g slow r e s p i r a t o r y j e t movements, the r e f i l l i n g phase i s powered completely by energy r e l e a s e d from the storage system ( G o s l i n e et al., 1983), and thus energy generated by the c o n t r a c t i o n of the c i r c u l a r muscles powers the e n t i r e j e t c y c l e . The cost of t h i s locomotion i s important, s i n c e t h i s mode i s used f o r normal steady swimming. U n l i k e the j e l l y f i s h s t u d i e d i n t h i s work, squ i d have r a d i a l muscles which can p a r t i a l l y , or completely power the r e f i l l i n g phase of the j e t c y c l e . These r a d i a l muscles are probably only a c t i v e d u r i n g r a p i d escape j e t s , when the a b s o l u t e c o s t of locomotion i s not of primary importance. The escape j e t i s used to a v o i d p r e d a t i o n , and i n c r e a s i n g the r a t e of the c y c l e f o r maximum a c c e l e r a t i o n would be most b e n e f i c i a l . The j e l l y f i s h s t u d i e d i n t h i s work maintain themselves i n the water column by c o n t r a c t i n g i n continous bouts of about 10-20 b e l l c o n t r a c t i o n s . These bouts of c o n t r a c t i o n s are analogous to the slow r e s p i r a t o r y j e t movements seen i n squid. As f o r the s q u i d , the c o s t s of t h i s locomotion are important, and the j e l l y f i s h would b e n e f i t by t a k i n g advantage of the 84 mechanisms d e s c r i b e d above. In f a c t , the s i g n i f i c a n c e of e l a s t i c s t r a i n energy storage systems i s remarkably d i s p l a y e d i n the j e l l y f i s h s t u d i e d i n t h i s paper. F i r s t l y , a n o n l i n e a r s t r e s s - s t r a i n curve was found f o r the locomotor s t r u c t u r e (Chapter I I ) , and thus these animals are c l e a r l y t a k i n g advantage of the mechanisms d e s c r i b e d above. That i s , the design of the energy storage system allows the same muscle to power both the c o n t r a c t i o n phase and the r e f i l l i n g phase of the c y c l e . Secondly, and q u i t e remarkably, these animals appear t o have adopted another important p h y s i o l o g i c a l c h a r a c t e r to take advantage of the n o n l i n e a r i t y of the energy storage system. Spencer and S a t t e r l i e (1981) found an unusual a c t i o n p o t e n t i a l i n the swimming muscle of P o l y o r c h i s . The a c t i o n p o t e n t i a l has a square waveform, which probably maintains the e x c i t a t i o n - c o n t r a c t i o n processes of the swimming muscles i n the a c t i v e s t a t e f o r long p e r i o d s of time. They suggest t h a t the f u n c t i o n a l s i g n i f i c a n c e of t h i s u n u s u a l l y l o n g a c t i o n p o t e n t i a l i s t h a t most of the water i n the subumbrellar c a v i t y can be e j e c t e d , and t h i s i s important i n g e n e r a t i n g l a r g e p r o p u l s i v e f o r c e s . I suggest that t h i s unusual a c t i o n p o t e n t i a l probably f u n c t i o n s not so much to generate l a r g e r hydrodynamic f o r c e s , as the p o t e n t i a l f o r g e n e r a t i n g such f o r c e s n e c e s s a r i l y decreases as the c o n t r a c t i o n c o n t i n u e s . Rather, i t all o w s s u f f i c i e n t deformation of the e l a s t i c s t r u c t u r e l a t e i n the c o n t r a c t i o n phase to s t o r e enough s t r a i n energy to antagonize the swimming muscles d u r i n g the recovery phase when the b e l l 85 expands. Spencer and S a t t e r l i e (1981) a l s o suggested that the d i s t i n c t p l a t e a u i n c a r d i a c muscle a c t i o n p o t e n t i a l may have the same f u n c t i o n a l s i g n i f i c a n c e they d e s c r i b e d f o r the a c t i o n p o t e n t i a l of the swimming muscle of P o l y o r c h i s . I suggest i t has the same f u n c t i o n a l s i g n i f i c a n c e d e s c r i b e d above. That i s , the extended a c t i o n p o t e n t i a l i n c a r d i a c muscle may a l l o w the muscles to deform e l a s t i c s t r u c t u r e s and s t o r e s t r a i n energy d u r i n g p e r i o d s when f u n c t i o n a l l y s i g n i f i c a n t hydrodynamic work: cannot be done. T h i s s t o r e d s t r a i n energy can then be used to h e l p r e s t o r e the deformed t i s s u e to r e s t i n g s t a t e s and a i d i n r e f i l l i n g the h e a r t . But t h i s would only be t r u e i f v e r t e b r a t e h e a r t s f u n c t i o n e d m e c h a n i c a l l y as s u c t i o n pumps, l i k e squid and j e l l y f i s h . In l i g h t of a recent new model proposed f o r the f u n c t i o n i n g of the v e r t e b r a t e h e a r t , where i t i s suggested that i n f a c t v e r t e b r a t e h e a r t s do f u n c t i o n as mechanical s u c t i o n pumps (Robinson et a l . , 1986), then the square waveform a c t i o n p o t e n t i a l of c a r d i a c muscle may w e l l have the mechanical f u n c t i o n d e s c r i b e d above. In Chapter II i t i s shown that the i s o l a t e d mesoglea comprising the system has a dynamic r e s i l e n c e of about 58%. T h i s i m p l i e s that a f t e r the r e l e a s e of the s t o r e d s t r a i n energy, w i t h i t s a s s o c i a t e d v i s c o u s l o s s e s , the o r i g i n a l energy -5 -5 w i l l be reduced to between 1.1 x 10 and 2.4 x 10 J of 86 energy l e f t t o power the r e f i l l i n g phase. The energy r e q u i r e d to power the r e f i l l i n g phase was shown to be between -5 -5 1.7 x 10 J and 2.1 x 10 J . These r e s u l t s v e r i f y t h a t the mesoglea can a c t as an e f f e c t i v e e l a s t i c s t r u c t u r e t h a t can completely power the r e f i l l i n g phase. I t i s tempting to suggest, i n l i g h t of the remarkable s i m i l i a r i t y i n magnitude between the a v a i l a b l e and r e q u i r e d energy f o r the r e f i l l i n g phase, that the storage system i s tuned to f u n c t i o n a t some optimum l e v e l . T h i s i d e a w i l l be e x p l o r e d i n d e t a i l i n the next chapter. 87 CHAPTER IV. THE PRESENCE AND IMPORTANCE OF A RESONANT PHENOMENON IN THE LOCOMOTOR STRUCTURE A.INTRODUCTION The phenomenon of resonance has profound importance throughout p h y s i c s . I t can be observed when a p h y s i c a l o s c i l l a t o r i s s u b j e c t e d to a p e r i o d i c d r i v i n g f o r c e by an e x t e r n a l agency. A p e r i o d i c f o r c e of f i x e d s i z e produces very d i f f e r e n t amplitudes of o s c i l l a t i o n , depending on i t s frequency. I f the d r i v i n g frequency i s a t , or near, the n a t u r a l frequency of the d r i v e n o s c i l l a t o r , then the amplitude of the o s c i l l a t i o n i s very l a r g e f o r repeated a p p l i c a t i o n s of a small f o r c e . D r i v i n g f r e q u e n c i e s above or below the n a t u r a l frequency of the o s c i l l a t o r produce comparatively small amplitudes of o s c i l l a t i o n f o r the same f o r c e . Locomotor apparatus i n metazoan animals are d r i v e n by muscles, the b i o l o g i c a l f o r c e g e n e r a t o r s , to o b t a i n the maximum amplitude of o s c i l l a t i o n , thus maximizing t h e i r mechanical f u n c t i o n t o p r o p e l the organism. The a b i l i t y to i n c r e a s e the f o r c e generated by muscle beyond some maximum v a l u e , however, i s l i m i t e d by i t s p h y s i o l o g y (Alexander, 1985). Animals might t h e r e f o r e d e s i g n locomotor s t r u c t u r e s that couple the frequency of a c t i v a t i o n of muscles w i t h the n a t u r a l , resonant frequency 88 of the locomotor s t r u c t u r e , thus t a k i n g advantage of the phenomenon of resonance. Animals working at the resonant frequency of the locomotor s t r u c t u r e c o u l d then o b t a i n a maximum amplitude of o s c i l l a t i o n i n the locomotor s t r u c t u r e w i t h the minimum mass of muscle t o f o r c e the o s c i l l a t o r , while keeping the maximum f o r c e generated constant. A locomotor s t r u c t u r e working a t or near resonance can thus a c t as an energy c o n s e r v i n g d e v i c e . T h i s i d e a i s not new. McMahon (1975, 1985) has proposed that hopping kangaroos and g a l l o p i n g quadrupeds have a s t r i d e frequency t h a t i s the resonant frequency of the body. T a y l o r (1985) suggests t h a t h i s recent experiments on human hopping shows that the r o l e of e l a s t i c s t r a i n energy i s maximized and metabolic energy i s minimized when s u b j e c t s hop a t t h e i r n a t u r a l frequency. T h i s chapter models the locomotor b e l l of the hydromedusean P o l y o r c h i s p e n i c i l l a t u s as a moderately damped harmonic o s c i l l a t o r , and shows that the p r e d i c t e d resonant frequency of the o s c i l l a t o r i s at or near the working frequency of the animal. The s t r u c t u r e of the b e l l has been d e s c r i b e d i n d e t a i l i n Chapter I, but the i n f o r m a t i o n i s summarized here. The swimming muscles l i n e the s u r f a c e of the subumbrellar mesoglea and are arranged to decrease the diameter of the b e l l d u r i n g c o n t r a c t i o n s . No muscles e x i s t to antagonize t h i s movement. The b e l l i t s e l f i s made of n o n c e l l u l a r mesoglea that i s t r a v e r s e d by numerous r a d i a l l y arranged f i b e r s . These 89 f i b e r s are loaded i n t e n s i o n d u r i n g c o n t r a c t i o n s of the b e l l , and rough c a l c u l a t i o n s d e s c r i b e d i n Chapter I I show t h a t there are probably enough of the f i b e r s present i n the s t r u c t u r e to s t o r e a l l the p o t e n t i a l s t r a i n energy r e q u i r e d to antagonize the c o n t r a c t i o n of the swimming muscles. There i s a l s o evidence presented i n Chapter I I I that the amount of energy r e q u i r e d to power the r e f i l l i n g , and the amount p r o v i d e d by the energy storage system are c l o s e l y matched, sugge s t i n g that the energy storage system i s tuned t o work a t some optimum. That i s , mechanical energy i s not wasted by s t o r i n g more energy i n the e l a s t i c storage system then i s needed t o j u s t power the r e f i l l i n g phase. 90 B. MATERIALS AND METHODS 1. Experimental L i v e P o l y o r c h i s were c o l l e c t e d i n Bamfield I n l e t , B r i t i s h Columbia, and maintained i n running seawater a q u a r i a . Free damped o s c i l l a t i o n s of the locomotor s t r u c t u r e of a s i n g l e animal were recorded from the f r e e o s c i l l a t i o n s f o l l o w i n g s i n g l e spontaneous c o n t r a c t i o n s of the swimming muscles. To r e c o r d these d a t a , a specimen was t e t h e r e d t o a f l a t P l e x i g l a s p l a t e at the apex of the b e l l (with c y a n o a c r y l a t e a d h e s i v e ) . The b e l l i s t r a n s p a r e n t , and i t i s p o s s i b l e to observe changes i n the i n t e r n a l dimensions of the animal d u r i n g c o n t r a c t i o n s of the b e l l . A v i d e o system was used t o monitor r e a l time changes i n the i n t e r n a l diameter of the t e t h e r e d animal. T h i s system uses a Video Dimension A n a l y z e r (Model 303, Instruments f o r Ph y s i o l o g y and Medicine, San Diego, CA) t h a t p r o v i d e s an e l e c t r i c a l s i g n a l which i s p r o p o r t i o n a l to the s e p a r a t i o n of two c o n t r a s t boundaries on any h o r i z o n t a l l i n e i n the vi d e o image. The f r e e damped o s c i l l a t i o n s were recorded on a Hewlett-Packard i n s t r u m e n t a t i o n tape r e c o r d e r (Model 3964A) and l a t e r p r i n t e d on a Hewlett-Packard c h a r t r e c o r d e r (Model 7402A) f o r f i n a l a n a l y s i s . Most of the animals examined d u r i n g these s t u d i e s d i d not 91 g i v e c l e a n , s i n g l e spontaneous c o n t r a c t i o n s f o l l o w e d by f r e e damped o s c i l l a t i o n s , as seen f o r the i n d i v i d u a l i n t h i s study. T h i s i n d i v i d u a l c o n t r a c t e d i n such a manner f o r the e n t i r e l e n g t h of these experiments, and allowed adequate time to set up the v i d e o system and to c o l l e c t v ery c l e a n records of the f r e e damped o s c i l l a t i o n s . I t u n f o r t u n a t e l y d i d not cooperate by c o n t r a c t i n g i n continuous t r a i n s d u r i n g any time of the experiment. The working frequency of eleven f r e e l y swimming animals was measured by a frame-by-frame a n a l y s i s of vi d e o records taken of other animals c o n t r a c t i n g i n continuous t r a i n s . 2. A n a l y t i c a l The locomotor s t r u c t u r e of the j e l l y f i s h was modelled as a damped harmonic o s c i l l a t o r (see f o r example Kleppner and Kolenkow, 1973). The equation of motion f o r such an o s c i l l a t o r i s : mX + bX + kX = 0 (4.1) where X i s the displacement, and i n t h i s experiment, i s the i n s i d e circumference of the b e l l . The other terms are the 92 f i r s t (X) and second d e r i v a t i v e (X) of the displacement, and re p r e s e n t , r e s p e c t i v e l y , the v e l o c i t y and a c c e l e r a t i o n of the body w a l l . Each of the p h y s i c a l constants a s s o c i a t e d w i t h the three terms i n the equation has a b i o l o g i c a l analogy. k, the s p r i n g c o n s t a n t , i s the s t r u c t u r a l s t i f f n e s s of the b e l l . b, the damping c o e f f i c i e n t , accounts f o r both the i n t e r n a l f r i c t i o n i n the t i s s u e i t s e l f , r e s u l t i n g from the v i s c o e l a s t i c p r o p e r t i e s of the mesoglea, and the e x t e r n a l f r i c t i o n i n the water r e s u l t i n g from the shear f o r c e s generated d u r i n g the f l o w of water out of and around the b e l l , m, the mass of the system, i s the e f f e c t i v e mass of the b e l l . I t i n c l u d e s both the a c t u a l mass of the animal and the mass of any water t h a t i s a c c e l e r a t e d by the c o n t r a c t i o n of the b e l l . Values f o r the three constants were obtained u s i n g separate methods. The s p r i n g constant, k, was obtained from an independent study (see below). The damping c o e f f i c i e n t , b, was measured from the f r e e damped o s c i l l a t i o n s recorded by methods d e s c r i b e d above. The mass of the o s c i l l a t o r , m, c o n t a i n i n g the b e l l i t s e l f and any water e n t r a i n e d d u r i n g the o s c i l l a t i o n was not known. But t h i s mass can be c a l c u l a t e d as f o l l o w s , from standard equations f o r damped o s c i l l a t o r s . I t can be shown that the l o g a r i t h m i c decrement, the r a t i o of s u c c e s s i v e maximum displacements of a f r e e damped o s c i l l a t o r i s d e f i n e d by: 93 l o g (X' /X" ) = b/4mf (4.2) e where X' and X'' are the displacements and f i s the c i r c u l a r frequency. These s u c c e s s i v e maximum data can a l s o he used to c a l c u l a t e the r e s i l e n c e of the o s c i l l a t o r per h a l f c y c l e as (Alexander, 1983): 2 R = (X''/X') (4.3) Now the angular frequency, uJ , f o r a damped harmonic o s c i l l a t o r can be d e f i n e d as: 2 2 2 Co = k/m - b /4m (4.4) S o l v i n g E q u a t i o n 4.2 f o r b and s u b s t i t u t i n g i n t o Equation 4.4 y i e l d s an equation d e f i n i n g the mass of the s p r i n g i n terms of e x p e r i m e n t a l l y measureable q u a n t i t i e s . 94 2 2 2 m = k /£ u; + 4f Clog ( X ' / X " ) 3 J (4.5) e The only unknown i s the s p r i n g c o n s t a n t , k. However, the dynamic s t r u c t u r a l s t i f f n e s s (E) f o r the swimming s t r u c t u r e , where the s t r a i n i s d e f i n e d i n terms of changes i n the i n s i d e circumference of the b e l l , i s shown i n Chapter I I to between -2 400 and 1000 N m . The s t r u c t u r a l s t i f f n e s s can be converted to a s p r i n g constant w i t h the f o l l o w i n g equation: k = L E t / T f ^ ( 4 . 6 ) where L i s the h e i g h t of the b e l l , t i s the r e s t i n g t h i c k n e s s of the b e l l , and r i s the r e s t i n g i n s i d e r a d i u s . The O d e r i v a t i o n of t h i s equation i s shown i n Appendix I I . These equations provide enough i n f o r m a t i o n to determine the equation of motion of the locomotor s t r u c t u r e , modelled as a harmonic o s c i l l a t o r . One c l e a r advantage of m o d e l l i n g the 95 locomotor s t r u c t u r e i n such a manner becomes evident here, s i n c e i t i s p o s s i b l e to p r e d i c t how the locomotor s t r u c t u r e w i l l respond t o an a p p l i e d d r i v i n g f o r c e of any g i v e n frequency. A f o r c e d damped harmonic o s c i l l a t o r has a frequency dependent amplitude of o s c i l l a t i o n and i s d e f i n e d by: 2 2 2 2 1/2 A( uJ ) = F 0 /mC( Ul0 - UJ ) + ( UJ t ) 3 (4.7) where F o i s the maximum amplitude of the s i n u s o i d a l d r i v i n g f o r c e and ^ 0 i s the n a t u r a l frequency of the undamped o s c i l l a t o r , d e f i n e d by: 1/2 = (Jc/m) (4.8) and 1$ i s d e f i n e d by: * = b/ m (4.9 96 The assumptions which were made i n order t o model the locomotor apparatus as a h a r m o n i c a l l y f o r c e d , damped o s c i l l a t o r cannot be v e r i f i e d e x p e r i m e n t a l l y . However, i t i s p o s s i b l e to make a s p e c i f i c p r e d i c t i o n u s i n g the model and compare the p r e d i c t i o n to e x p e r i m e n t a l l y measured v a l u e s of the same q u a n t i t y . The p r e d i c t i o n made was the work done per c y c l e by the e x c i t i n g f o r c e . The d e r i v a t i o n of the equation used to make the p r e d i c t i o n i s summarized i n Appendix I I I . The equation used was: 2 2 W = 1 T F 0 A s i n C a r c t a n ( w/ u; 0 - Cu ) D (4.10) A l l symbols i n t h i s equation have been d e s c r i b e d p r e v i o u s l y . 97 C. RESULTS M o d e l l i n g any system as a h a r m o n i c a l l y d r i v e n damped o s c i l l a t o r r e q u i r e s a measure of three parameters: (1) the s t i f f n e s s of the s p r i n g p r o v i d i n g the e l a s t i c r e s t o r i n g f o r c e , (2) the damping i n the system and (3) the mass of the o s c i l l a t o r . The s p r i n g i n the locomotor system of t h i s animal i s the mesogleal e l a s t i c s t r u c t u r e c o n t a i n i n g a system of r a d i a l ' e l a s t i c ' f i b e r s . C e r t a i n morphometric data, and the estimate of the dynamic s t r u c t u r a l s t i f f n e s s of the b e l l taken from Chapter I I can be used w i t h E q u a t i o n 4.6 to c a l c u l a t e the s p r i n g constant of the e l a s t i c system of the locomotor apparatus. The necessary morphometric data from the animal used to measure the f r e e damped o s c i l l a t i o n s i n c l u d e : H, the -2 h e i g h t of the b e l l , 2.0 x 10 m; t , the r e s t i n g t h i c k n e s s of -2 the b e l l , .25 x 10 m; r , the r e s t i n g i n s i d e r a d i u s of the -2 subumbrellar c a v i t y , .86 x 10 m. These morphometric data and the dynamic s t r u c t u r a l s t i f f n e s s e s d e f i n e d above s u b s t i t u t e d i n t o E quation 4.6 y i e l d a s p r i n g constant of between 0.74 and -1 1.85 N m . The measurement of the damping parameters can be made d i r e c t l y from records of f r e e damped o s c i l l a t i o n s . A t y p i c a l f r e e damped o s c i l l a t i o n of the i n s i d e diameter of the b e l l i s shown i n F i g u r e 4.1. E i g h t o s c i l l a t i o n s were recorded. 98 F i g u r e 4.1. A t y p i c a l f r e e v i b r a t i o n of the i n s i d e diameter of the b e l l of the hydromedusean P o l y o r c h i s . The h o r i z o n t a l bar i s one second, and the maximum amplitude of o s c i l l a t i o n i s 6.46 mm. The shaded area shows o s c i l l a t i o n s where the p r e s s u r e changes i n the subumbrellar c a v i t y are expected to be l a r g e . 99 100 C a r e f u l i n s p e c t i o n of F i g u r e 4.1 shows that both the f r e q u e n c i e s of o s c i l l a t i o n s and the r e l a t i v e amplitudes of s u c c e s s i v e o s c i l l a t i o n s d i f f e r between the f i r s t o s c i l l a t i o n (shaded) and the subsequent o s c i l l a t i o n s . T h i s suggests t h a t two d i s t i n c t processes are o c c u r r i n g . These processes become apparent by i n s p e c t i o n of the pressure changes t h a t occur d u r i n g the j e t c y c l e . Simultaneous measurement of pressure changes i n the subumbrellar c a v i t y and f r e e damped o s c i l l a t i o n s of the i n t e r n a l diameter of the subumbrellar c a v i t y of t h i s animal were not made. Pressure-diameter records from the previous Chapter can be used to i n f e r what processes are o c c u r r i n g d u r i n g the f r e e o s c i l l a t i o n s . F i g u r e s 3.1 and 3.2 show t h a t p r e s s u r e s are l a r g e d u r i n g the f o r c e d c o n t r a c t i o n but decay s i g n i f i c a n t l y d u r i n g the r e f i l l i n g . I would expect pressure changes d u r i n g the o s c i l l a t i o n s t h a t f o l l o w the f o r c e d o s c i l l a t i o n of F i g u r e 4.1 to be very s m a l l . Thus f l u i d f l o w w i l l be l a r g e d u r i n g the f i r s t o s c i l l a t i o n (shaded), but comparatively small d u r i n g the f r e e damped o s c i l l a t i o n s t h a t f o l l o w . T h i s suggests that d u r i n g the i n i t i a l f o r c e d c o n t r a c t i o n i n F i g u r e 4.1 the measured damping terms w i l l r e f l e c t a complex i n t e r a c t i o n r e s u l t i n g from both the f r i c t i o n a l l o s s e s a s s o c i a t e d w i t h both the flow out of the subumbrellar c a v i t y and around the e x t e r i o r of the b e l l and w i t h mechanical h y s t e r e s i s a s s o c i a t e d w i t h the deformation of the b e l l mesoglea i t s e l f . The damping terms 101 measured from the second and subsequent o s c i l l a t i o n s w i l l mostly r e f l e c t the mechanical h y s t e r e s i s a s s o c i a t e d w i t h the deformation of the b e l l mesoglea. Thus data from the f i r s t o s c i l l a t i o n (shaded) w i l l be used to estimate the parameters used i n the model. Data from the second o s c i l l a t i o n w i l l be used as an independent measurement of the m a t e r i a l p r o p e r t i e s of the b e l l mesoglea. Data taken from the f i r s t o s c i l l a t i o n of a l l e i g h t damped o s c i l l a t i o n s , as i l l u s t r a t e d i n F i g u r e 4.1, are summarized i n Table 4.1. A l l r e p o r t e d parameters t h a t were measured from these records are averaged v a l u e s . The e r r o r s a s s o c i a t e d w i t h these measurements are standard d e v i a t i o n s of the mean v a l u e s . The r a t i o of these s u c c e s s i v e amplitudes (X /X ) can be 1 2 s u b s t i t u t e d i n t o E quation 4.2 to g i v e the l o g a r i t h m i c decrement. E i g h t of such measurements y i e l d an average locrarithmic decrement of 1.078. The c i r c u l a r frequency of the -1 f i r s t o s c i l l a t i o n i s 1.17 c y c l e s sec , corresponding to an -1 angular frequency of 7.35 ra d i a n s sec These data can be used i n Eq u a t i o n 4.5 to c a l c u l a t e the e f f e c t i v e mass of the o s c i l l a t o r . The numerator on the r i g h t s i d e of the equation i s the s t i f f n e s s of the s p r i n g , and i s -1 between 0.74 and 1.85 N m . S u b s t i t u t i o n of these values i n t o E q uation 4.5 g i v e s an e f f e c t i v e mass of between 0.0123 and 0.0306 kg. For comparison, the r e a l wet mass of the e n t i r e 102 TABLE 4.1 Numerical v a l u e s of the parameters estimated from F i g u r e 4.1 f o r the f i r s t o s c i l l a t i o n (shaded) or d e r i v e d from the equations (Values i n p a r e n t h e s i s are standard d e v i a t i o n s of the means (n=8)). Parameter Value U n i t s " f 1.17 (.028) Hz -1 ^ 7.35 (.15) rads sec 1.078 ( .073) m .0123 - .0306 kg -1 k .74 - 1.85 N m -1 b .0618 - .154 N m sec 103 animal was 0.005 kg, and the mass of the e n t i r e animal p l u s the mass of the water contained i n the r e s t i n g subumbrellar cavity-i s about 0.011 kg. These data can be used t o measure the damping parameter, b, used i n the equation of motion of the o s c i l l a t o r . S o l v i n g Equation 4.2 f o r b, and s u b s t i t u t i o n of the a p p r o p r i a t e v a l u e s y i e l d s a damping parameter of between 0.062 and 0.154 -1 N m sec. These val u e s can be used w i t h Equation 4.7 to p r e d i c t the frequency dependence of the amplitude of o s c i l l a t i o n f o r a s i n u s i o d i a l l y v a r y i n g f o r c e w i t h some constant maximum amplitude. F i g u r e 4.2 shows t h i s frequency dependence f o r a harmonic o s c i l l a t o r w i t h the parameters d e f i n e d above f o r the i e l l y f i s h . The maximum amplitude has been normalized to the maximum amplitude measured from the f r e e damped o s c i l l a t i o n s from F i g u r e 4.2. For a constant maximum f o r c e , the maximum amplitude of o s c i l l a t i o n i s reached a t a -1 frequency of about 7 ra d i a n s sec . T h i s corresponds to a c i r c u l a r frequency of about 1.1 Hz. T h i s i s the p r e d i c t e d resonant frequency of the locomotor system f o r t h i s p a r t i c u l a r animal. I t should be noted t h a t v a r i a t i o n i n the dynamic s t r u c t u r a l s t i f f n e s s measured i n Chapter II th a t i s propogated i n t o t h i s Chapter i s e l i m i n a t e d i n the c a l c u l a t i o n s used to generate F i g u r e 4.2. The range of v a l u e s measured f o r the 104 F i g u r e 4.2. The dependence of the r e l a t i v e amplitude of o s c i l l a t i o n f o r v a r i o u s f r e q u e n c i e s of a constant maximum amplitude of the e x c i t i n g f o r c e . The maximum p r e d i c t e d amplitude was normalized to the maximum amplitude of o s c i l l a t i o n measured i n F i g u r e 4.1. 105 106 dynamic s t r u c t u r a l s t i f f n e s s was used to set a range of values on the s p r i n g constant used i n the model i n t h i s Chapter. T h i s range of v a l u e s f o r the s p r i n g constant was then used to set a range of v a l u e s on the damping parameter (Equation 4.2) and the e f f e c t i v e mass (Equation 4.5). I n s p e c t i o n of Equations 4.2 and 4.5 shows t h a t these q u a n t i t i e s are d i f f e r e n t by constant m u l t i p l i e r s . T h e r e f o r e the same frequency dependence, as shown i n F i g u r e 4.2, i s obtained when any value of the s p r i n g constant i s used i n the c a l c u l a t i o n s . Continuous t r a i n s of c o n t r a c t i o n s were not observed i n t h i s animal, thus the n a t u r a l swimming frequency f o r t h i s p a r t i c u l a r animal was not known. However, F i g u r e 4.3 shows the waveform of continuous c o n t r a c t i o n s f o r two other t e t h e r e d animals measured u s i n g the v i d e o system d e s c r i b e d above. The working f r e q u e n c i e s are about 0.8 Hz and 1.0 Hz. Free swimming animals have a mean frequency of 1.1 Hz (S=.43) when c o n t r a c t i n g i n continuous t r a i n s . Data f o r the f r e e o s c i l l a t i o n s f o l l o w i n g the f o r c e d o s c i l l a t i o n can be used to estimate m a t e r i a l p r o p e r t i e s of the locomotor apparatus. The r a t i o of s u c c e s s i v e maximum amplitudes can be used to c a l c u l a t e the r e s i l i e n c e of the m a t e r i a l . The r a t i o X /X measured from a l l e i g h t records of 3 2 the f r e e damped o s c i l l a t i o n s s u b s t i t u t e d i n t o E q u a t i o n 4.3 g i v e a mean value of 61% f o r the r e s i l e n c e of the mesogleal material. 108 F i g u r e 4.3. T y p i c a l records of the i n s i d e diameter of the b e l l of a j e l l y f i s h c o n t r a c t i n g i n continuous t r a i n s . The h o r i z o n t a l s c a l e bar i s one second. 109 1 1 0 D. DISCUSSION The mesogleal b e l l of the j e l l y f i s h Polyorchis was modelled as a damped, harmonically forced o s c i l l a t o r . Four major assumptions were made i n making th i s model. F i r s t l y , c l a s s i c a l harmonic o s c i l l a t o r s are modelled as massless springs with a known mass attached i n ser ies at one end of the spr ing . The ' spr ing ' i n the j e l l y f i s h i s the b e l l mesoglea i t s e l f . The ' spr ing ' i s therefore not massless, and the attached mass of the o s c i l l a t o r becomes, i n p a r t , the spring i t s e l f , as wel l as the mass of the rest of the b e l l , and the mass of water entrained by the movement. Secondly, and somewhat re la ted to the f i r s t assumption, a l inear s t r e s s - s t r a i n curve i s assumed to ex is t for the mechanical propert ies of the e l a s t i c s tructure . T h i r d l y , the damping term i n the equation of motion, which includes both the i n t e r n a l damping of the t i ssue and the damping due to f r i c t i o n a l losses within the moving f l u i d , i s assumed to be proport ional to the v e l o c i t y of movement, thus ignoring any higher order terms. It should be noted that the assumption of l inear damping great ly s i m p l i f i e s the so lut ion of Equation 4.1, since i f higher order terms were included i n the damping term, a so lut ion of Equation 4.1 could only be obtained through numerical methods. This assumption i s reasonably v a l i d . I l l however. F i r s t l y , the i n t e r n a l damping of the t i s s u e i s n e a r l y independent of the v e l o c i t y of the deformation. T h i s can be seen i n F i g u r e 2.4 where the damping f a c t o r s are reasonably independent of the frequency of the measurement. The apparatus used to measure the damping f a c t o r s e n f o r c e d a deformation a t a r a t e that i s d i r e c t l y p r o p o r t i o n a l to the frequency of the deformation. Secondly, the damping due to f r i c t i o n a l l o s s e s w i t h i n the moving f l u i d i s probably p r o p o r t i o n a l to the v e l o c i t y t o some power l e s s than two. T h i s can be seen from data presented i n t h i s Chapter. Data from F i g u r e 4.1 can be used to estimate the Reynolds number f o r the movement of the bodv w a l l d u r i n g the c o n t r a c t i o n phase. The v e l o c i t y of the -1 movement of the b e l l i s about 1 cm sec . Since the b e l l h e i g h t of the animal i s about 2 cm, the Reynolds number ( v e l o c i t y x height/.01 - see B a t c h e l o r , 1977) i s about 200. In the range of Reynolds numbers of between about 10 to 1000 damping changes from a p u r e l y l i n e a r f u n c t i o n of v e l o c i t y to a f u n c t i o n of v e l o c i t y squared (see f o r i n s t a n c e B a t c h e l o r , 1977). Thus i t seems l i k e l y t h a t damping i n P o l y o r c h i s i s p r o p o r t i o n a l t o some power l e s s than 2. F o u r t h l y , when p r e d i c t i n g how the o s c i l l a t o r w i l l respond to an a p p l i e d d r i v i n g f o r c e of any g i v e n frequency the d r i v i n g f o r c e i s assumed to be s i n u s o i d a l . The f o r c e generators i n t h i s o s c i l l a t o r are the swimming muscles, and these do not generate t e n s i o n s i n u s o i d a l l y (Spencer and S a t t e r l i e , 1981). 112 however, the r e s u l t i n g displacement of the locomotor s t r u c t u r e i s v e ry n e a r l y a s i n u s o i d a l f u n c t i o n (see F i g u r e 4.3). The v a l i d i t y of these assumptions i s d i f f i c u l t t o t e s t , but a s p e c i f i c p r e d i c t i o n made by the model i s i n e x c e l l e n t agreement w i t h independent measurements made e x p e r i m e n t a l l y i n Chapter I I I . T h i s i m p l i e s t h a t the model i s robust and means that i t s v a l i d i t y i s not s e r i o u s l y a f f e c t e d by moderate d e v i a t i o n s from the u n d e r l y i n g assumptions. The robustness of the model was t e s t e d by u s i n g Equation 4.10 to p r e d i c t the frequency dependent work output per c y c l e f o r the o s c i l l a t o r . T h i s can be compared d i r e c t l y to completely independent data c o l l e c t e d i n Chapter I I I . Morphometric data used i n Equation 4.10 were taken from the animal used i n Chapter I I I . The necessary data a r e : the -2 heicrht of the b e l l , 2.7 x 10 m; the r e s t i n g i n s i d e r a d i u s of -2 the subumbrellar c a v i t y , 1.0 x 10 m; the r e s t i n g t h i c k n e s s -2 of the b e l l , 0.25 x 10 m. These d a t a , a l o n g w i t h the parameters measured from F i g u r e 4.1, were used i n the equations d e s c r i b e d i n the M a t e r i a l s and Methods to c a l c u l a t e a l l the parameters i n Equ a t i o n 4.10. The amplitude of the o s c i l l a t i o n was taken from F i g u r e 3.1 and i s the change i n the circumference of the b e l l . I t was c a l c u l a t e d from one h a l f of -3 the peak to peak change i n diameter, and i s 7.9 x 10 m. F measured from E q u a t i o n AIII.8 has a maximum value of about 0.01 N. The angular frequency used was the measured frequency 113 -1 from F i g u r e 3.1, 5 rads sec . These c a l c u l a t i o n s y i e l d a work -5 -4 output per c y c l e of between 7.0 x 10 J and 1.8 x 10 J . T h i s r e p r e s e n t s the energy d i s s i p i a t e d d u r i n g an e n t i r e c y c l e . For a h a l f c y c l e , the p r e d i c t e d energy d i s s i p i a t e d w i l l be -5 -5 between 3.5 x 10 J and 7.8 x 10 J . T h i s can be compared to data from Chapter I I I , where the energy d i s s i p i a t e d f o r the c o n t r a c t i o n phase was measured e x p e r i m e n t a l l y . The energy d i s s i p i a t e d w i l l be equal to the energy d i s s i p i a t e d i n g e n e r a t i n g the pressure i n the -5 subumbrellar c a v i t y , 5.4 x 10 J (see Table 3.1) p l u s the energy d i s s i p i a t e d i n deforming the t i s s u e . The l a t t e r v a lue cannot be seen d i r e c t l y i n Table 3.1. but can be c a l c u l a t e d from the o r i g i n a l data i n Chapter I I . From the l o s s modulus of the mesogleal m a t e r i a l , the energy d i s s i p i a t e d d u r i n g the -5 -5 deformation i s between 0.4 x 10 J and 0.9 x 10 J . The energy d i s s i p i a t e d d u r i n g the c o n t r a c t i o n phase of the c y c l e , -5 -5 t h e r e f o r e , i s between 5.8 x 10 and 6.3 x 10 J . The p r e d i c t e d energy d i s s i p i a t e d f o r a h a l f c y c l e n i c e l y spans the valu e s measured i n independent experiments, im p l y i n g t h a t the model i s robust; t h e r e f o r e any moderate v i o l a t i o n s i n the assumptions s t a t e d p r e v i o u s l y w i l l not i n v a l i d a t e the model. I t should be noted t h a t these data can be used as d e s c r i b e d i n Chapter I I I to compare w i t h p r e v i o u s estimates ( D a n i e l , 1985; Bone and Trueman, 1982) of the power 114 requirements of l e t p r o p u l s i o n . Using the methods d e s c r i b e d i n Chapter I I I . the measured power requirements f o r P o l y o r c h i s are -5/3 between 0.27 and 0.29 W kg . These c a l c u l a t i o n s a l s o agree w i t h D a n i e l ' s estimates where the va l u e s range from 0.2 to 0.75 -5/3 W kg . Bone and Trueman (1982) measured power requirements -5 -4 of between 3.24 x 10 and 1.35 x 10 W f o r A b y l o p s i s and -4 -4 between 3.88 x 10 and 7.76 x 10 W f o r C h e l o p h y s i s . These a l s o agree reasonably w e l l w i t h the estimates f o r P o l y o r c h i s -5 -5 (5.8 x 10 and 6.3 x 10 W). The maximum muscle f o r c e p r e d i c t e d by Equation AIII.8 used i n the equations to p r e d i c t the work done per c y c l e by the animal i n Chapter I I I i s 0.01 N. T h i s can be compared to the maximum f o r c e measured e x p e r i m e n t a l l y f o r the same animal. The maximum s t r e s s generated by the c o n t r a c t i o n of the muscle i s 5 -2 shown i n Chapter I I I to be at l e a s t 1.25 x 10 N m . The c r o s s - s e c t i o n a l area of the muscles i s approximately -8 2 5.4 x 10 m . M u l t i p l i c a t i o n of these two valu e s g i v e s a measured f o r c e of about 0.007 N. The measured and p r e d i c t e d values of the maximum value of the f o r c e generated by the c o n t r a c t i o n of the muscles are i n good agreement. Bone and Trueman (1982) d i d not pro v i d e the necessary morphometric data to c a l c u l a t e muscle f o r c e s f o r the j e l l y f i s h they examined. Trueman (1980) gave a va l u e of 0.02 N f o r the f o r c e generated by a 130 g j e l l y f i s h . He d i d not d e s c r i b e the methods used to c a l c u l a t e t h i s f o r c e . 115 The p r e d i c t e d resonant frequency of the locomotor s t r u c t u r e f o r the animal used to c o l l e c t the damping parameters i s 1.1 Hz. The working frequency, i e . the frequency of c o n t r a c t i o n f o r an animal c o n t r a c t i n g i n a continuous t r a i n , was not measured f o r the animal d e s c r i b e d here. However, the mean value of the working frequency f o r e i g h t free-swimming animals i s 1.1 Hz. T h i s f o r t u i t o u s r e s u l t c e r t a i n l y suggests that a l l P o l y o r c h i s swim at t h e i r r e s p e c t i v e resonant f r e q u e n c i e s . T h i s does not imply, however, t h a t a l l these animals swim at e x a c t l y t h a t p a r t i c u l a r frequency. In f a c t , v a r i a t i o n s i n t h i s working frequency c e r t a i n l y e x i s t . For i n s t a n c e , the animals swimming i n the t r a i n s of c o n t r a c t i o n s shown i n F i g u r e 4.3 have working f r e q u e n c i e s of about 0.8 Hz and 1 Hz. I f the animal w i t h the lower working frequency had a p r e d i c t e d frequency dependent amplitude of o s c i l l a t i o n as shown i n F i g u r e 4.2, i t would work at a frequency w e l l t o the l e f t of the peak amplitude. But i n d i v i d u a l v a r i a t i o n s i n the s t r u c t u r a l s t i f f n e s s of the b e l l , the damping parameters, and the s i z e of the animals c o u l d generate an e n t i r e spectrum of resonant curves. I suggest, t h e r e f o r e , t h a t each i n d i v i d u a l , when c o n t r a c t i n g i n continuous t r a i n s , i s f u n c t i o n i n g a t , or near, the resonant frequency of i t s locomotor apparatus. What does the animal g a i n by f o r c i n g the locomotor s t r u c t u r e a t i t s resonant frequency? T a y l o r (1985) suggests, i n human hopping a t l e a s t , that the r o l e of e l a s t i c energy 116 storage Is maximized and that metabolic energy consumption i s minimized when s u b j e c t s hop a t t h e i r n a t u r a l frequency. I d i d not c a r r y out metabolic s t u d i e s on t h i s animal, so I cannot address the q u e s t i o n of p o t e n t i a l metabolic savings i n c u r r e d by t h i s animal working a t resonance. However, data from Chapter II suggest t h a t there i s a c l o s e matching between the q u a n t i t y of e l a s t i c s t r a i n energy s t o r e d i n the s p r i n g system, and the amount of energy needed from t h i s storage system to power the -5 r e f i l l i n a phase. I t i s shown t h a t between 1.0 x 10 and -5 2.4 x 10 J of energy w i l l be a v a i l a b l e from the energy o r i g i n a l l y s t o r e d i n the system. Other mechanical measurements -5 -5 show that between 1.7 x 10 and 2.1 x 10 J of energy would be r e q u i r e d to power the r e f i l l i n g phase. The animal that was examined was c o n t r a c t i n g i n continuous t r a i n s , and presumably at or near the resonant frequency of i t s locomotor system. I t would seem then, t h a t one advantage of working a t resonance i s that the animals can maximize the use of t h e i r storage system. Thus, mechanical energy that had been, d i v e r t e d from the metabolic energy of the c o n t r a c t i o n of the muscles w i l l not be wasted by an i n e f f i c i e n t storage system. Only that energy t h a t i s r e q u i r e d t o power the r e f i l l i n g w i l l be d i v e r t e d from other mechanical e n e r g i e s . F i g u r e 4.2 shows t h a t f o r a constant maximum f o r c e , the animals can i n c r e a s e the amplitude of the o s c i l l a t i o n s by about 40% above the amplitude f o r s i n g l e c o n t r a c t i o n s , i f they f o r c e 117 the b e l l a t the resonant frequency. Increases i n the amplitude of o s c i l l a t i o n are important f o r j e t - p r o p e l l e d animals, s i n c e l a r g e r volumes of water can be e x p e l l e d f o r the same maximum f o r c e . I t may, however, be more i n s t r u c t i v e to examine the work done by the muscles when they are f o r c i n g the b e l l a t and o f f i t s resonant frequency. Equation 4.10 can be used to c a l c u l a t e the work done by the e x c i t i n g f o r c e f o r any frequency. I t would seem a p p r o p r i a t e then to use E q u a t i o n 4.10 to p r e d i c t the work done near the resonant frequency, and f a r from the resonant frequency, and then compare the magnitudes of the p r e d i c t e d v a l u e s . T h i s comparison, however, cannot be made u s i n g Equation 4.10. The p h y s i c a l model e n f o r c e s a change i n the r a t e of the deformation of the b e l l as the angular frequency v a r i e s . T h i s comparison, t h e r e f o r e , would be q u i t e a r b i t r a r y , s i n c e p r e d i c t e d changes i n the work would mostly r e f l e c t changes i n the d i s s i p a t i v e f o r c e s c r e a t e d by changes i n the r a t e s of both deformation of the body w a l l and f l u i d flow. A more p h y s i o l o g i c a l l y r e l e v a n t comparison would be to compare the p r e d i c t e d work done by an animal swimming at resonance to the work done i n completing a s i n g l e c o n t r a c t i o n a t the same r a t e of deformation without resonance. The work done by the muscles f o r a s i n g l e c o n t r a c t i o n , but at the same r a t e of deformation cannot be measured d i r e c t l y . 118 Estimates of t h i s q u a n t i t y can be made by a more d e t a i l e d a n a l y s i s of data presented i n the pr e v i o u s Chapter. In that chapter, the mechanical energy generated by the muscles i s e x p e r i m e n t a l l y measured, and as a f i r s t approximation, these energ i e s are a l g e b r a i c a l l y summed to g i v e the t o t a l energy generated d u r i n g the c o n t r a c t i o n . An examination of energy changes t h a t occur i n an o s c i l l a t o r working a t , or o f f , i t s resonant frequency leads d i r e c t l y to an a l t e r n a t i v e method of a n a l y z i n g those data. For any o s c i l l a t o r f u n c t i o n i n g a t resonance, such as a simple mass on a s p r i n g , p o t e n t i a l and k i n e t i c e n e r g i e s f l u c t u a t e s i n u s o i d a l l y between the energy s t o r e d i n the s p r i n g , and the energy a s s o c i a t e d w i t h the movement of the mass of the o s c i l l a t o r . At any p o i n t i n time, the sum of the k i n e t i c and p o t e n t i a l e n e r g i e s i s constant. That i s , as the s p r i n g i s compressed near the extremes of displacement of the o s c i l l a t o r , i t s t o r e s p o t e n t i a l energy that has been t r a n f e r r e d from the k i n e t i c energy a s s o c i a t e d w i t h the movement of the o s c i l l a t o r . A l l of the energy i n the system i s s t o r e d as p o t e n t i a l energy i n the s p r i n g . Near the e q u i l i b r i u m p o s i t i o n of the displacement, most of the energy has been t r a n s f e r r e d i n t o k i n e t i c energy, and l i t t l e energy i s s t o r e d i n the s p r i n g . At resonance, the e x t e r n a l f o r c e o n l y does work to counter d i s s i p a t i v e p r o c e s s e s . T h i s can be seen q u a l i t a t i v e l y w i t h a c l o s e r examination of data i n Table 3.1. I f the animal i s 119 working a t resonance, the k i n e t i c and p o t e n t i a l energies should be about equal i n magnitude. T h i s i s r e f l e c t e d i n the approximate equal magnitudes of the energy r e q u i r e d to deform the t i s s u e and the energy a s s o c i a t e d w i t h the i n e r t i a of the w a l l , summed f o r the c o n t r a c t i o n and r e f i l l i n g phase. The energy the swimming muscles generate, t h e r e f o r e , i f the swimming muscles i n t h i s animal are f o r c i n g the b e l l t o work at it's resonant frequency, can be approximated as the energy to overcome the d i s s i p a t i v e f o r c e s . T h i s energy i s shown above to -5 -5 be between 5.8 x 10 and 6.3 x 10 J . The energy to overcome the d i s s i p a t i v e f o r c e s d u r i n g the r e f i l l i n g should not be i n c l u d e d i n t h i s summation, s i n c e the swimming muscles do not d i r e c t l y do t h i s work. The r e f i l l i n g phase i s powered by the r e l e a s e of energy from the deformation of the t i s s u e . Off resonance, the k i n e t i c and p o t e n t i a l e n e r g i e s are not t r a n s f e r r e d from one to the o t h e r , and t h e i r sum i s not constant at any time. For example below resonance where e l a s t i c f o r c e s dominate, the energy generated by the f o r c e can be approximated as the energy to counter the d i s s i p a t i v e f o r c e s and the energy r e q u i r e d to deform the s p r i n g . T h i s can be measured from data presented i n Table 3.1. The enercry r e q u i r e d -5 to counter the d i s s i p a t i v e f o r c e s i s between 5.8 x 10 and -5 6.3 x 10 J . The energy r e q u i r e d t o deform the t i s s u e i s -5 -5 between 1.8 x 10 and 4.1 x 10 J . T h e r e f o r e o f f resonance, the energy generated by the swimming muscles i s between 120 -5 -4 7.6 x 10 and 1.0 x 10 J . A comparison of these e n e r g i e s t o -5 the e n e r g i e s generated at resonance (5.8 x 10 and -5 6.3 x 10 J) g i v e s an estimate of the p o t e n t i a l energy savings of working at resonance. I f the swimming muscles of these animals f o r c e the b e l l at a frequency t h a t i s near the resonant frequency of the b e l l , the e n e r g e t i c requirements f o r the c y c l e w i l l be reduced by about 24% to 37% of the t o t a l energy generated by the swimming muscles. The advantages of f o r c i n g the s t r u c t u r e at i t s resonant frequency are t h e r e f o r e q u i t e remarkable. F u n c t i o n i n g at resonance may a l s o p r o v i d e some advantages to the p h y s i o l o g y of the swimming muscles. I d i d not examine the temporal r e l a t i o n s h i p s of the movement of the b e l l and the c o n t r a c t i o n of the muscles. But i f the muscles i n i t i a t e d a c o n t r a c t i o n near the end of the r e f i l l i n g phase, the i n e r t i a of the w a l l c o u l d a c t i v e l y s t r e t c h the muscles. The muscles t h e r e f o r e c o u l d p r o v i d e a s t r a i n energy storage system t o i n i t i a t e the f o l l o w i n g c o n t r a c t i o n . The c o n t r a c t i o n of muscles a f t e r a c t i v e s t r e t c h i n g has been shown to g r e a t l y i n c r e a s e the e f f i c i e n c y and t o t a l work output of muscles (Heglund and Cavagna, 1985). T h i s i d e a , t h a t r e s o n a n t i n g s t r u c t u r e s use the i n e r t i a of the rebounding o s c i l l a t o r to s t r e t c h a c t i v e muscles i s i n t r i g u i n g , and c e r t a i n l y warrants f u r t h e r i n v e s t i g a t i o n . Other organisms t h a t have r e s o n a t i n g locomotor systems may i n i t i a t e muscle c o n t r a c t i o n s so t h a t they are a c t i v e l y 121 s t r e t c h e d w h i l e a c t i v e , thus i n c r e a s i n g the e f f i c i e n c y of muscle c o n t r a c t i o n . I have not examined s c a l i n g phenomenon r e l a t e d to resonance i n t h i s animal, although s i z e dependent v a r i a t i o n i n the resonant f r e q u e n c i e s would be expected because mass i s one of the parameters c h a r a t e r i z i n g an o s c i l l a t o r . I t i s very i n t e r e s t i n g t h at one aspect of the p h y s i o l o g y of the swimming muscles, however, seems f u n c t i o n a l l y r e l a t e d to the s c a l i n g of the resonant f r e q u e n c i e s . A p o s i t i v e c o r r e l a t i o n between the s i z e of the animal (measured as the b e l l diameter) and the d u r a t i o n of the t e n s i o n development has been shown to e x i s t (Spencer and S a t t e r l i e , 1981). I f the model presented here i s v a l i d , then i t i s tempting to i n f e r t h a t the s i z e dependence on the d u r a t i o n of the t e n s i o n development i s an attempt to a d j u s t the f o r c i n g frequency to accommodate f o r n a t u r a l , r e q u i s i t e changes i n the resonant frequency of the locomotor s t r u c t u r e as the animal grows. Another p h y s i o l o g i c a l phenomenon p o t e n t i a l l y r e l a t e d t o the i d e a of resonance i s that the swimming muscles have a r e f r a c t o r y p e r i o d t h a t e x a c t l y matches the swimming frequency (Spencer, p e r s o n a l communication, 1986). T h i s would i n s u r e t h a t the muscles w i l l not c o n t r a c t a t f r e q u e n c i e s above the resonant frequency. F i n a l l y , the l o g a r i t h m i c decrement c a l c u l a t e d f o r the f i r s t o s c i l l a t i o n can be compared to s i m i l i a r data c o l l e c t e d 122 p r e v i o u s l y ( D a n i e l , 1985). D a n i e l was unable to d i r e c t l y measure the damped o s c i l l a t i o n s as the instantaneous change i n the i n s i d e diameter of the b e l l as measured i n t h i s work. He estimated the damping from volume changes t h a t o c c u r r e d i n the subumbrellar c a v i t y d u r i n g spontaneous c o n t r a c t i o n s of the swimming muscles. He measured the volume changes from a frame-by-frame a n a l y s i s of f r e e swimming animals f i l m e d w i t h a movie camera. Data g i v e n i n h i s F i g u r e 2 g i v e a l o g a r i t h m i c decrement of about 1.9, as compared to the value of about 1.1 c a l c u l a t e d f o r t h i s work. The damping measured w i t h both of these methods accounts f o r both the i n t e r n a l f r i c t i o n i n the t i s s u e i t s e l f , r e s u l t i n g from the v i s c o e l a s t i c p r o p e r t i e s of the mesoglea, and the e x t e r n a l f r i c t i o n i n the water r e s u l t i n g from the shear f o r c e s generated d u r i n g the f l o w of water out of and around the b e l l . D i f f e r e n c e s i n the anatomy of the mesoglea i n the locomotor s t r u c t u r e s of the s p e c i e s compared might a f f e c t the f i r s t component, and d i f f e r e n c e s i n the shape of the b e l l might i n f l u e n c e the second component, s i n c e the f l o w regimes of water out and around the b e l l would be d i f f e r e n t . D i f f e r e n c e s i n the anatomy of the mesoglea and the shape of the b e l l of the s p e c i e s examined by D a n i e l (Gonionemus  ve r t e n s and Stomotoca a t r a ) and i n t h i s work e x i s t . G l a d f e l t e r (1973) shows t h a t the two s p e c i e s of hydromedusea examined by D a n i e l do not c o n t a i n the c h a r a c t e r i s t i c j o i n t s of s o f t mesoglea. D i f f e r e n c e s i n the shapes of the b e l l s are c l e a r l y seen i n A r a i and Brinckmann-Voss (1980). I t i s not s u r p r i s i n g 123 that d i f f e r e n c e s i n the measured l o g a r i t h m i c decrements e x i s t . No mention has been made y e t to the data c o l l e c t e d f o r the second and f o l l o w i n g o s c i l l a t i o n s seen i n the f r e e damped o s c i l l a t i o n s of F i g u r e 4.1. These data are unimportant t o the mechanics of the locomotor system, e s p e c i a l l y w i t h regards to the phenomenon of resonance. But these data can be used to examine the m a t e r i a l p r o p e r t i e s of the locomotor apparatus. Because pr e s s u r e s are very small i n the subumbrellar c a v i t y d u r i n g these o s c i l l a t i o n s , water movement i n and around the b e l l would be expected to be minimal. Thus shear s t r e s s e s i n the water w i l l be s m a l l , but c e r t a i n l y not zero. Any measure of damping w i l l mostly r e f l e c t v i s c o u s damping i n the mesoglea, and data can be used to measure the p r o p e r t i e s of the m a t e r i a l comprising the mesoglea i t s e l f . E q uation 4.3 was used to c a l c u l a t e the r e s i l e n c e of the m a t e r i a l , and the mean value c a l c u l a t e d was 61%. T h i s can be compared t o the r e s i l e n c e of i s o l a t e d mesoglea measured i n dynamic t e s t s i n Chapter I I . The average r e s i l e n c e of the i s o l a t e d mesoglea i s 58%. These r e s u l t s are i n e x c e l l e n t agreement, and suggest that i s o l a t i n g the mesoglea f o r the dynamic t e s t s d i d not a l t e r i t s mechanical p r o p e r t i e s . 124 CHAPTER V. SUMMARY I t appears t h a t when the hydromedusean j e l l y f i s h P o l y o r c h i s p e n i c i l l a t u s swims i n continuous bouts, i t i s probably f o r c i n g i t s b e l l to o s c i l l a t e a t the resonant frequency of the s t r u c t u r e . Quite remarkably, t h i s behaviour w i l l save between 24% and 37% of the t o t a l energy r e q u i r e d to generate the j e t t h a t p r o p e l s the animal, and a l l o w s the animal to i n c r e a s e the amplitude of the o s c i l l a t i o n by about 40%, as compared to a s i n g l e c o n t r a c t i o n o f f resonance. T h i s work presents the data on the mechanics of the j e t c y c l e necessary to v e r i f y t h i s i d e a . 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B i o l . 58, 567-583. Johnson, W., Soden, P.D., Trueman, E.R. (1973). A study i n j e t p r o p u l s i o n : an a n a l y s i s of the motion of the squid L o l i g o v u l g a r i s . J . exp. B i o l . 56, 155-165. Kleppner, D. and Kolenkow, R.J. (1973). An I n t r o d u c t i o n to Mechanics. McGraw-Hill Book Company, Toronto. Koehl, M.A.R. (1977a). Mechanical d i v e r s i t y of connective t i s s u e of the body w a l l of sea anemones. J . Exp. B i o l . 69, 107-125. Koehl, M.A.R. (1977b). Mechanical o r g a n i z a t i o n of 130 c a n t i l e v e r - l i k e organisms: sea anemones. J . Exp. B i o l . 69, 127-142. Love, A.E.H. (1944). A T r e a t i e s on the Mathematical Theory of E l a s t i c i t y . 4th E d i t i o n . New York: Dover P u b l i c a t i o n s . McDonald, D.A. (1974). Blood Flow i n A r t e r i e s . Second E d i t i o n . Edward A r n o l d , London. McMahon, T.A. (1975). Using body s i z e t o understand the s t r u c t u r a l d e s i g n of animals: quadrupedal locomotion. J . a p p l . P h y s i o l . 39, 619-627. McMahon, T.A. (1985). The r o l e of compliance i n mammalian running g a i t s . J . exp. B i o l . 115, 263-282. Moore, J.D. and Trueman, E.R. (1971). Swimming of the s c a l l o p , Chiamys o p e r c u l a r i s ( L . ) . J . exp. mar. B i o l . E c o l . 6, 179-185. Packard, A. (1969). J e t p r o p u l s i o n and the g i a n t f i b e r response of L o l i g o . Nature, Lond. 221, 875-877. Robinson, T.F., F a c t o r , S.M. and Sonnenblick, E.H. (1986). The h e a r t as a s u c t i o n pump. S c i . Amer. 254(6), 131 84-93. Shadwick, R.E. and G o s l i n e , J.M. • (1985). Mechanical p r o p e r t i e s of the octopus a r t e r y . J . Exp. B i o l . 114, 259-284. Spencer, A.N. and S a t t e r l i e , R.A. (1981). The a c t i o n p o t e n t i a l and c o n t r a c t i o n i n the subumbrellar swimming muscle of P o l o r c h i s p e n i c i l l a t u s (Hydromedusae). J . Comp. P h y s i o l . 144: 401-407. T a y l o r , C.R. (1985). Force development d u r i n g s u s t a i n e d locomotion: a determinant of g a i t , speed and metabolic power. J . exp. B i o l . 115, 253-262. Trueman, E.R. (1980). Swimming by j e t p r o p u l s i o n . Aspects of Animal Movement, (eds H.Y. E l d e r and E.R. Trueman), pp. 93-105. Cambridge: Cambridge U n i v e r s i t y P r e s s . Wainwright, S.A., Biggs, W.D., Currey, J.D., and G o s l i n e , J.M. (1976). Mechanical Design i n Organisms. London: Edward A r n o l d . Zar, J.H. (1984). B i o s t a t i s t i c a l A n a l y s i s . 2nd E d i t i o n . New J e r s e y : P r e n t i c e - H a l l , Inc. 132 APPENDIX I. T h i s appendix d e s c r i b e s the c a l c u l a t i o n s used to convert the s t a t i c pressure-volume data to s t r e s s - s t r a i n d a ta. In order to convert the pressure-volume data to s t r e s s - s t r a i n data two assumptions were made. (1) The mesoglea i s a constant volume t i s s u e , thus knowing t h a t the j e l l y f i s h - p i p e t system was c l o s e d , then any volume change i n the p i p e t must equal changes i n the volume of the subumbrellar c a v i t y . (2) The b e l l c a v i t y i s approximately c y l i n d r i c a l w i t h an e s s e n t i a l l y c i r c u l a r c r o s s - s e c t i o n ( G l a d f e l t e r , 1972). S t r a i n was d e f i n e d i n terms of changes i n the i n s i d e r a d i u s of the b e l l : S t r a i n =(R - R WR ( A I . l ) i n o , i n o , i n where R = f i n a l i n s i d e r a d i u s and R = i n i t i a l i n s i d e i n o , i n r a d i u s . T h i s d e f i n i t i o n of s t r a i n i s commonly c a l l e d 'engineering s t r a i n ' , and i s an approximation good only at small s t r a i n s . For l a r g e s t r a i n s (above about 10%) i t i s more a p p r o p r i a t e to d e f i n e s t r a i n as t r u e s t r a i n (see Wainwright et a l . , 1976). The s t r a i n was d e f i n e d here as e n g i n e e r i n g s t r a i n t o be c o n s i s t e n t w i t h the d e f i n i t i o n of s t r a i n used i n the model of Chapter IV, where the s t r a i n s d e r i v e d from Equation 133 A I . l are used t o c a l c u l a t e a s p r i n g constant t h a t i t s e l f uses e n g i n e e r i n g s t r a i n i n i t s d e f i n i t i o n . I t can be shown that d i f f e r e n c e s i n the parameters c a l c u l a t e d u s i n g s t r a i n d e f i n e d as e n g i n e e r i n g or true s t r a i n are w i t h i n the experimental e r r o r s of the measurments. C i r c u m f e r e n t i a l s t r e s s was d e f i n e d as i n Love (1944) f o r a t h i c k w a l l e d c y l i n d e r , but m o d i f i e d below t o c a l c u l a t e s t r e s s at the i n s i d e r a d i u s and assuming zero p r e s s u r e at the o u t s i d e r a d i u s , y i e l d i n g : ( 2 ) ( 2 ) ( P * R ) + ( p A R ) ( i n in) ( i n out) S t r e s s = (AI.2) 2 2 R -R out i n where R i s the f i n a l o u t s i d e r a d i u s , P i s the i n s i d e out p r e s s u r e , and R i s d e f i n e d as above. In order to c a l c u l a t e i n s t r e s s and s t r a i n as d e f i n e d above, g e o m e t r i c a l r e l a t i o n s h i p s were d e r i v e d to d e f i n e the i n s i d e and o u t s i d e r a d i u s of the j e l l y f i s h a f t e r a known change i n volume, as f o l l o w s . The circumference of the b e l l was d e f i n e d as twice the b e l l width; t h e r e f o r e , the i n i t i a l o u t s i d e r a d i u s of the b e l l i s : R = W/TT (AI.3) o ,out where W i s the b e l l width. Using g e o m e t r i c a l r e l a t i o n s h i p s and 134 Equation AI.3, the i n i t i a l volume of the j e l l y f i s h can be d e f i n e d as: 2 V = (W * H ) / Tr* (AI.4) o where H i s the b e l l h e i g h t . The above equation can be used to d e f i n e the b e l l width and b e l l h e i g h t f o r a known change i n volume, and w i t h Equation AI.3,'the o u t s i d e r a d i u s of the b e l l , a f t e r a change i n volume, becomes: 2 1/2 R = (W - CV * TT*/H 3) / TT' (AI.5) out where W and H are the i n i t i a l b e l l width and h e i g h t r e s p e c t i v e l y , V i s the volume of f l u i d removed. From g e o m e t r i c a l r e l a t i o n s h i p s , the i n i t i a l c r o s s - s e c t i o n a l area of the mesoglea i s : 2 2 A = IV * £ R - (R - T ) } (AI.6) o o,out o,out o where R i s d e f i n e d by Equation AI.3 and T i s the i n i t i a l o,out o t h i c k n e s s of the mesoglea. Since the c r o s s - s e c t i o n a l area remains a constant, then Equation AI.6 must be t r u e f o r any combination of R and T. S o l v i n g f o r T y i e l d s : 135 2 1/2 T = R - £ R - A/TT3 (AI.7) out out where R i s d e f i n e d by Equation AI.5 and A by Equation AI.6, out p r o v i d i n g an eq u a t i o n f o r the t h i c k n e s s of the mesoglea a f t e r a known change i n volume. The f i n a l i n s i d e r a d i u s becomes: R = R - T (AI.8; i n out where R and T are d e f i n e d by Equation AI.5 and Equation AI.7, out r e s p e c t i v e l y . 136 APPENDIX II This appendix describes the d e r i v a t i o n of Equation 4.6. It i s necessary to convert the dynamic s t r u c t u r a l s t i f fness measured i n Chapter II into a spring constant, k, for use i n the equations of motion of damped harmonic o s c i l l a t o r s of Chapter IV. The c ircumferent ia l stress for a th in walled cy l inder i s derived by d i v i d i n g the force ac t ing on the wal l of the cy l inder by the area of the wal l of the cy l inder (see Gordon, 1978, p. 120). The force i s defined by: F = 2PRH ( A I I . l where P i s the ins ide pressure, R i s the mean rad ius , and t i s the wal l thickness . The area i s defined as: 2tH ( A l l . 2 ) The c i rcumferent ia l stress then becomes: 137 6~ = PR/t ( A l l . 3 ) The s t r a i n was d e f i n e d as the r e l a t i v e change i n the circumference of the i n s i d e of the h e l l . £ = X/ 2VTr0 ( A l l . 4 ) where r 0 i s the r e s t i n g i n s i d e r a d i u s . By d i v i d i n g E quation A l l . 3 hy A l l . 4 to g i v e the s t i f f n e s s (E) and rearrangement, the change i n circumference becomes: X = 2TTPRr c /Et ( A l l . 5 ) The s p r i n g constant can be d e f i n e d by the r a t i o of F/x, or Equation A I I . l d i v i d e d by Equation A l l . 5 . S i m p l i f y i n g g i v e s : k = HEt / 1Y r 0 ( A l l . 6 ) 138 APPENDIX I I I . This appendix describes the der iva t ion of Equation 4.10 (see Hansen and Chenea, pp. 96-97, 1952). It estimates the work done per cycle by the locomotor muscles contract ing at any frequency. The d r i v i n g force was assumed to o s c i l l a t e s i n u s o i d a l l y , and can be defined by: F( t ) = F cos(CJt) ( A I I I . l ) o where F i s the maximum magnitude of the force , UJ i s the angular frequency of the contrac t ion , and t i s time. The t o t a l work done by such a f o r c e , over one period of o s c i l l a t i o n (T) , i s defined by the d e f i n i t e i n t e g r a l : X C T ) W= J (F 4 cosWt)dx (AIII.2) Xfo) which can be changed to a time i n t e g r a l and becomes: 139 W= / C(F 0 cos Wt)dx/dt3dt (AIII.3) For any s i n u s o i d a l l y d r i v e n o s c i l l a t o r , the displacement of the mass i s d e f i n e d by: X = A c o s ( u ; t + d> ) (AIII.4) where A i s the maximum amplitude of o s c i l l a t i o n , and d> i s the phase s h i f t between the a p p l i e d f o r c e and the displacement. T h i s f u n c t i o n , and i t s f i r s t d e r i v a t i v e , s u b s t i t u t e d i n t o Equation AIII.3 y i e l d s : T W = J -F A ^ ( c o s w t ) [ s i n ( w ; t + <£> >3dt (AIII.5) o With t r i g o n o m e t r i c i d e n t i t i e s , t h i s equation can be transformed i n t o the f o l l o w i n g equation: T W = j -F A v A J L " . 5 s i n ( 2 u ; t + < & > - . 5 s i n ( $ )3dt (AIII.6 o 140 Since .5 s i n ( 2 u J t + <I> )dt i s equal to zero, E q u a t i o n AIII.6 s i m p l i f i e s t o : W = F T f A s i n ( <Z> ) (AIII.7) The maximum f o r c e , F 0 , can be d e f i n e d by: F o = A ( c o 0 ) m K K (AIII.8) and d> by: 2 2 cb = a r c t a n ( J u; / ( W c - to )) (AIII.9) Publications of M. Edwin DeMont M.E. DeMont and J.M. Gosline. 1986. A natural resonating bell; the presence and importance of a resonant phenomenon in the locomotor structure of a hydromedusean jellyfish. Bulletin of the Canadian Society of Zoologists. Vol. 17 (2): p. 20. and Proceedings of the Pacific Division, American Association for the Advancement of Science. Vol. 5 (1): p. 27. (Abstracts) J.M. Gosline, M.E. DeMont and M.W. Denny. 1986. The structure and properties of spider silk. Endeavour. New Series. 10 (1): pp. 37-43. J.M. Gosline and M.E. DeMont. 1985. Jet-propelled swimming in squid. Scientific American. 252(1): pp. 96-103. J.M. Gosline, M.W. Denny and M.E. DeMont. 1984. Spider silk as rubber. Nature (London). 309(5968): pp. 551-552. M.E. DeMont and R.K. O'Dor. 1984. The effects of activity, temperature and mass on the respiratory metabolism of the squid, Illex illecebrosus. Journal of the Mraine Biological Association of the U.K. 64: pp. 535-543. J.M. Gosline, J.D. Steeves, A.D. Harman and M.E. DeMont. 1983. Patterns of circular and radial mantle muscle activity in respiration and jetting of the squid Loligo opalescens. Journal of Experimental Biology. 104. pp. 97-109. J.M. Gosline, R.E Shadwick and M.E. DeMont. 1982. Elastic energy storage in squid jetting. American Zooligist. Vol. 22(4): p. 941. (Abstract). M.E. DeMont. 1982. Maintenance of adult squid in captivity. Bulletin of the Canadian Society of Zoologists. Vol. 13(2): p. 33. (Abstract). R.W.M. Hirtle, M.E. DeMont and R.K. O'Dor. 1981. Feeding, growth and metabolic rates in captive short-finned squid, Illex illecebrosus in relation to the natural population. Journal of Shellfish Research. Vol 1(2): pp. 187-192. M.E. DeMont and R.K. O'Dor. 1981. Metabolic scope in the squid, Illex  illecebrosus. Bulletin of the Canadian Society of Zoologists. Vol 12(2): p. 47. (Abstract). 

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