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Stochastic Growth Patterns Generated by Phycomyces Sporangiophores

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R. IGOR GAMOW and DAVID E. CLOUGH University of Colorado, Department of Chemical Engineering, Boulder, CO 80309

The giant a e r i a l sporangiophore of the fungus Phycomyces blakesleeanus r o t a t e s as it elongates t r a c i n g out a left-handed s p i r a l . A number of years ago our laboratory became f a s c i n a t e d with the nature of t h i s spiral growth f o r it appeared to us that it must a c c u r a t e l y r e f l e c t the molecular a r c h i t e c t u r e of the growing plant cell w a l l . Fine s t r u c t u r e k i n e t i c a n a l y s i s of both the r o t a t i o n a l and the e l o n g a t i o n a l components of s p i r a l growth revealed what appeared to be random growth irregularities. We now show by using time s e r i e s a n a l y s i s that the apparent r o t a t i o n and elongation i r r e g u l a r i t i e s are t r u l y s t o c h a s t i c and, most s u r p r i s i n g l y , are not c o r r e l a t e d . D u r i n g t h e p a s t 100 y e a r s a number o f l a b o r a t o r i e s h a v e s t u d i e d the growing " a n t i c s " of the g i a n t , s i n g l e - c e l l e d sporan g i o p h o r e o f t h e f i l a m e n t o u s f u n g u s , Phycomyces b l a k e s l e e a n u s . " A n t i c s " i s a p p r o p r i a t e h e r e s i n c e , i n terms o f d e s c r i b i n g t h e v a r i e t y o f growth p a t t e r n s t h a t t h e mature s p o r a n g i o p h o r e p r o d u c e s , i t c e r t a i n l y a p p e a r s t o be t h e " c l o w n " o f t h e p l a n t w o r l d . Even l e f t t o i t s own d e v i c e s and n o t s u b j e c t e d t o any s e n s o r y i n p u t s , i t i s happy i n a d a r k , damp room a t a b o u t 22°C; i t s b e h a v i o r i s s t i l l q u i t e e x t r a o r d i n a r y b e c a u s e o f i t s phenomenal g r o w t h r a t e o f t h r e e m i l l i m e t e r s p e r h o u r (50 ym/min) s i m u l t a n e o u s l y c o u p l e d w i t h a r o t a t i o n r a t e o f 12 t o 15 d e g r e e s p e r m i n . Shown on t h e TV m o n i t o r i n F i g u r e 1 i s t h e u p p e r p a r t o f a m a t u r e Phycomyces s p o r a n g i o p h o r e c o n s i s t i n g o f a c y l i n d r i c a l s t a l k , a b o u t 100 ym i n d i a m e t e r crowned w i t h a s p h e r i c a l s p o r e s a c k , a b o u t 500 pm i n d i a m e t e r . B i o p h y s i c i s t s have l o n g b e e n f a s c i n a t e d w i t h t h i s o r g a n i s m because of i t s unique sensory a p p a r a t u s . Both the d i r e c t i o n o f g r o w t h and t h e r a t e o f g r o w t h a r e q u a n t i t a t i v e l y c o n t r o l l e d by a v a r i e t y of w e l l - d e f i n e d s t i m u l i : they respond t o both the i n t e n s i t y and t h e d i r e c t i o n o f e i t h e r v i s i b l e o r UV l i g h t , t o a v a r i e t y o f w i n d g r a d i e n t s , and t o t o u c h . The o r g a n i s m e l i c i t s c o m p l e x 0097-6156/83/0207-0403$06.00/0 ©

1983 American Chemical Society

Blanch et al.; Foundations of Biochemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1983.


Figure 1.

The video recorder and TV monitor of the video microscope. On the screen of the monitor is shown a mature stage IVb P h y c o m y c e s sporangiophore.

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o l f a c t o r y r e s p o n s e s , i t grows away f r o m g r a v i t y , and f i n a l l y , i t senses the presence o f d i s t a n t o b j e c t s and thus a v o i d growing i n t o them. D e t a i l s c o n c e r n i n g t h e s e b e h a v i o r s h a v e b e e n d e s c r i b e d i n a number o f r e v i e w a r t i c l e s ( 1 - 4 ) . The l a s t b e h a v i o r l i s t e d , t h e a v o i d a n c e r e s p o n s e , a p p e a r s t o be a c o m b i n a t i o n o f two i n d e p e n d e n t s e n s o r y i n p u t s , o l f a c t i o n and w i n d (_5, 6 ) . As a r e s u l t o f a l l these studies concerning t h e sensory apparatus, more i s known a b o u t t h e t w o - d i m e n s i o n a l g r o w t h p a t t e r n s p r o d u c e d by t h e Phycomyces s p o r a n g i o p h o r e t h a n a n y o t h e r p l a n t s y s t e m . I t i s t h u s a l l t h e more s u r p r i s i n g t h a t t h e d e t a i l e d f i n e s t r u c t u r e of t h e s p i r a l g r o w t h i s o n l y now b e g i n n i n g t o b e known. I n g e n e r a l , t h e o l d e r Phycomyces l i t e r a t u r e c o n s i d e r e d s p i r a l g r o w t h t o be a n a l o g o u s t o a r o t a t i n g s c r e w , i . e . t h e f a s t e r t h e s c r e w w o u l d r o t a t e , t h e f a s t e r i t would e l o n g a t e , and t o t h e f i r s t approxima tion this i s true. B u t d e t a i l e d f i n e s t r u c t u r e measurements o f r o t a t i o n and e l o n g a t i o n r a t e s w i t h i n t h e growing zone c l e a r l y showed t h a t t h i s a n a l o g y i s a p o o r one. Cohen a n d D e l b r i i c k ( 7 ) i n 1958 f i r s t r e a l i z e d t h a t a m a t e r i a l s e c t i o n o f t h e g r o w i n g z o n e , GZ, i s n o t e q u i v a l e n t t o t h e e n t i r e GZ b u t e a c h s e c t i o n o f t h e GZ i s i n s t e a d y s t a t e f l u x c h a r a c t e r i s t i c o f t h a t p a r t i c u l a r section. They a l s o f o u n d t h a t t h e d e g r e e o f r o t a t i o n and e l o n g a t i o n was n o t u n i f o r m l y d i s t r i b u t e d w i t h i n t h e GZ. I n 1974 O r t e g a , H a r r i s a n d Gamow (8) d e t e r m i n e d t h e r a t i o o f r o t a t i o n t o e l o n g a t i o n a s a f u n c t i o n o f g r o w i n g zone p o s i t i o n a n d f o u n d t h a t t h i s r a t i o i n c r e a s e d a s measurements w e r e t a k e n t o w a r d s t h e l o w e r edge o f t h e g r o w i n g z o n e . M o s t r e c e n t l y , Gamow and B o t t g e r ( 9 ) , u s i n g e i t h e r h i g h r e s o l u t i o n 35 mm p h o t o g r a p h y o r a v i d e o TV m i c r o s c o p e , h a v e f o u n d t h a t when t h e m i n u t e - p e r - m i n u t e r o t a t i o n a l a n d e l o n g a t i o n a l growth r a t e s were s i m u l t a n e o u s l y measured, b o t h r a t e s were q u i t e i r r e g u l a r and " a p p e a r e d " t o b e random. I n a d d i t i o n , i t " a p p e a r e d " t h a t l i t t l e o r no c o r r e l a t i o n e x i s t e d b e t w e e n t h e i r r e g u l a r i t i e s i n t h e r o t a t i o n r a t e and t h e e l o n g a t i o n r a t e . I n t h i s p a p e r we h a v e t r e a t e d t h e s e g r o w t h d a t a , b o t h r o t a t i o n a n d e l o n g a t i o n , t o what i s p o p u l a r l y known a s t i m e s e r i e s a n a l y s i s . We h a v e c o n c l u d e d f r o m t h e s e a n a l y s e s t h a t n o t o n l y a r e t h e m e a s u r e d i r r e g u l a r i t i e s i n r o t a t i o n and e l o n g a t i o n p u r e l y random white n o i s e , b u t that they a r e , i n a d d i t i o n , n o t c o r r e l a t e d . We would expect t h a t on a m i c r o s c o p i c s c a l e , a l l growth p r o c e s s e s w o u l d b e s t o c h a s t i c , b u t what i s s u r p r i s i n g i s t h a t t h e m a c r o s c o p i c g r o w t h k i n e t i c s o f a Phycomyces s p o r a n g i o p h o r e i s a l s o stochastic. M a t e r i a l s and M e t h o d s W i l d - t y p e Phycomyces b l a k e s l e e a n u s s p o r a n g i o p h o r e s , N R R L 1 5 5 5 ( - ) , o r i g i n a l l y o b t a i n e d f r o m M. D e l b r i i c k , w e r e grown i n s h e l l v i a l s c o n t a i n i n g 5.0% p o t a t o d e x t r o s e a g a r (PDA) w i t h 1.0% yeast extract. The s h e l l v i a l s were i n c u b a t e d u n d e r d i f f u s e i n c a n d e s c e n t l i g h t i n a h i g h - h u m i d i t y room w i t h a t e m p e r a t u r e r a n g e


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b e t w e e n 22° and 27°C. B e f o r e e a c h e x p e r i m e n t t h e s p o r a n g i o p h o r e s were d a r k a d a p t e d i n r e d l i g h t f o r a t l e a s t 20 m i n u t e s . A l l ex p e r i m e n t s were c a r r i e d out w i t h a w a t e r - f i l t e r e d r e d l i g h t source. The a p p a r a t u s u s e d t o s i m u l t a n e o u s l y measure m i n u t e - b y - m i n u t e t h e n e t r o t a t i o n and t h e n e t e l o n g a t i o n o f a s t a g e IVb GZ was f i r s t d e s c r i b e d by Cohen and D e l b r u c k (7) and t h e n m o d i f i e d by Gamow and B o t t g e r ( 9 ) . The m a t u r e s t a g e IVb s p o r a n g i o p h o r e i n a g l a s s s h e l l v i a l was f i r m l y s e c u r e d t o a s t a g e t h a t r o t a t e d c l o c k w i s e once e v e r y 60 s e c o n d s . To e n s u r e t h a t t h e GZ o f t h e s p o r a n g i o p h o r e was v e r t i c a l ( p a r a l l e l t o t h e a x i s o f r o t a t i o n o f t h e s t a g e ) , a d o u b l e knee was i n s e r t e d b e t w e e n t h e s t a g e and t h e vial. The r o t a t i n g s t a g e a l l o w e d us t o measure t h e a n g u l a r v e l o c i t y o f any p a r t i c l e s i t u a t e d a b o v e , b e l o w , o r i n t h e GZ. Be c a u s e t h e n e t r o t a t i o n o f a m a t u r e s t a g e IVb s p o r a n g i o p h o r e i s i n t h e same d i r e c t i o n a s t h e r o t a t i n g s t a g e ( c l o c k w i s e ) , a p a r t i c l e e i t h e r above o r i n t h e GZ t a k e s l e s s t h a n 60 s e c o n d s t o c o m p l e t e one r e v o l u t i o n . By d e t e r m i n i n g how much l e s s , we c a l c u l a t e d t h e a n g u l a r v e l o c i t y of the p a r t i c l e i n r e s p e c t t o the o b s e r v e r . (For i n s t a n c e , i f a p a r t i c l e c o m p l e t e s one r e v o l u t i o n i n 58 s e c o n d s , we c a n e a s i l y c a l c u l a t e t h a t t h e e n t i r e r e g i o n b e l o w t h e p a r t i c l e must h a v e a t o t a l a n g u l a r v e l o c i t y o f 12°/58 s e c o r 12.4°/min.) F o r o u r p r e s e n t e x p e r i m e n t s , we u s e d a g l a s s b e a d a p p r o x i m a t e l y 15 pm i n d i a m e t e r . The b e a d was p l a c e d on t h e s t a l k a p p r o x i m a t e l y 50 urn b e l o w t h e s p o r a n g i u m ; t h i s r e g i o n o f t h e s t a l k shows no e l o n g a t i o n . The GZ, w i t h t h e a t t a c h e d g l a s s b e a d , was o b s e r v e d c o n t i n u o u s l y t h r o u g h a TV v i d e o camera a t t a c h e d t o a l o w - p o w e r e d (^X10 m i c r o s c o p e ) . An e l e c t r o n i c t i m e r w i t h a d i g i t a l p r i n t o u t r e c o r d s t h e t i m e w i t h a r e s o l u t i o n o f 10 milliseconds. Experimental Controls. The r o t a t i o n a l v e l o c i t y o f a p a r t i c l e p l a c e d j u s t b e l o w t h e GZ was m e a s u r e d i n o r d e r t o d e t e r m i n e t h e r o t a t i o n measurement e r r o r . V e l o c i t y measurements o f a p a r t i c l e b e l o w t h e GZ make a n e x c e l l e n t c o n t r o l s i n c e a l l t h e param e t e r s a r e t h e same e x c e p t t h e i n n a t e r o t a t i o n o f t h e GZ i t s e l f . We m e a s u r e d t h e r o t a t i o n r a t e f o r s u c h a p a r t i c l e o n c e e v e r y m i n u t e f o r 56 c o n s e c u t i v e m i n u t e s and d e t e r m i n e d t h a t t h e s t a n d a r d d e v i a t i o n was ±1.7° p e r m i n u t e ( 9 ) . The b e a u t y o f o u r r o t a t i o n m e a s u r i n g method i s t h a t t h e measurements a r e v i r t u a l l y i n s e n s i t i v e t o p a r a l l a x p r o b l e m s and t o t h e h u n t i n g p r o b l e m m e n t i o n e d below. C a l c u l a t i n g e l o n g a t i o n a l v e l o c i t i e s , on t h e o t h e r h a n d , i s b e s e t w i t h numerous d i f f i c u l t i e s . Although the photographs are not n e c e s s a r y i n o r d e r t o c a l c u l a t e the a n g u l a r v e l o c i t i e s , t h e y a r e needed i n o r d e r t o measure t h e e l o n g a t i o n a l v e l o c i t i e s . To d e t e r m i n e o u r e r r o r f o r t h e e l o n g a t i o n a l measurement, a n a r t i f i c i a l s p o r a n g i o p h o r e was c o n s t r u c t e d . The a r t i f i c i a l s p o r a n g i o p h o r e c o n s i s t e d o f a s t r a i g h t 0.5 mm d i a m e t e r w i r e a t t a c h e d t o a m o t o r - d r i v e n micrometer screw. A s m a l l s p u r g e a r on t h e m o t o r s h a f t o f a s y n c h r o n o u s c l o c k m o t o r (1 rpm) d r i v e s a l a r g e r s p u r g e a r on a m e t r i c m i c r o m e t e r head w i t h a g e a r r a t i o o f 1:8. The


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m i c r o m e t e r s c r e w was a d v a n c i n g a t 63 pm/min. The a r t i f i c i a l P h y c o was p h o t o g r a p h e d n e x t t o a c a l i b r a t e d s c a l e p l a c e d i n t h e o c u l a r of the microscope. From a s e r i e s o f m i n u t e - p e r - m i n u t e p h o t o g r a p h s , we d e t e r m i n e d t h a t t h e s t a n d a r d d e v i a t i o n o f o u r e r r o r was ±1.4 ym p e r m i n ( 9 ) . T h i s p h o t o g r a p h i c method i s h i g h l y d e p e n d e n t on u s i n g s t r a i g h t s p o r a n g i o p h o r e s t o a v o i d parallax errors. P e r f e c t l y straight-growing sporangiophores are n e a r l y i m p o s s i b l e t o o b t a i n s i n c e t h e y show b o t h f a s t , 5 t o 7.5 m i n , and s l o w , 30 t o 60 m i n o s c i l l a t i o n s ; a t e r m u s e d t o d e s c r i b e these o s c i l l a t i o n s i s h u n t i n g (10). I n order to v e r i f y t h a t our measurements w e r e n o t s i g n i f i c a n t l y i n f l u e n c e d by e i t h e r a p a r a l l a x p r o b l e m o r s p o r a n g i o p h o r e h u n t i n g , we p l a c e d an a d d i t i o n a l m a r k e r b e l o w t h e g r o w i n g z o n e and c a l c u l a t e d d i r e c t l y t h e change i n l e n g t h b e t w e e n t h e two m a r k e r s . The d a t a o b t a i n e d i n t h i s manner w e r e no d i f f e r e n t f r o m d a t a o b t a i n e d u s i n g t h e c a l i b r a t e d s c a l e w i t h r e a s o n a b l y s t r a i g h t s p o r a n g i o p h o r e s . From a s e r i e s of p h o t o g r a p h s t a k e n w i t h a o n e - m i n u t e i n t e r v a l , s m a l l bends w e r e s e e n t o o c c u r i n t h e GZ, b u t s i n c e t h e bend a n g l e b e t w e e n any p a i r o f p h o t o g r a p h s was l e s s t h a n a d e g r e e , t h e e r r o r i n t e r m s of bending i s n e g l i g i b l e . We h a v e d e t e r m i n e d t h a t h u n t i n g c a u s e s a b e n d i n g r a t e o f a b o u t 0.5 d e g r e e / m i n , and i n a c o n t i n u o u s s e t of measurements l a s t i n g f o r 38 m i n u t e s , t h e b e n d i n g a n g l e w i l l n o t v a r y more t h a n f i v e d e g r e e s f r o m t h e v e r t i c a l . From a s i n g l e p a i r o f p h o t o g r a p h s , we c a n n o t e l i m i n a t e t h e p o s s i b i l i t y t h a t t h e grow i n g z o n e i s b e n t d i r e c t l y t o w a r d s o r d i r e c t l y away f r o m t h e cam e r a , b u t s i n c e we f i n d no s i g n i f i c a n t bends f r o m a l a r g e random s a m p l e o f p h o t o g r a p h s , we h a v e d i s c o u n t e d t h i s p o s s i b i l i t y . A TV v i d e o camera i n c o n j u n c t i o n w i t h a TV m o n i t o r and v i d e o c a s s e t t e r e c o r d e r ( F i g u r e 1) h a s an enormous a d v a n t a g e o v e r t h e c o n v e n t i o n a l 35 mm p h o t o g r a p h y . F i r s t l y , s i n c e an e n t i r e e x p e r i ment c a n be s t o r e d on a TV c a s s e t t e and t h u s becomes p a r t o f a permanent Phycomyces l i b r a r y , we c a n r e r u n any g i v e n e x p e r i m e n t , many l a s t i n g s e v e r a l h o u r s , a t any f u t u r e d a t e . S e c o n d l y , t h e r e l i a b i l i t y o f measurements i s enhanced s i n c e r e p l i c a t e s may be t a k e n and t h e t a p e d e x p e r i m e n t may be s t o p p e d o r rewound t o c h e c k results. E x p e r i m e n t a l R e s u l t s and S t o c h a s t i c M o d e l i n g . M e a s u r e m e n t s o f r o t a t i o n a l and e l o n g a t i o n a l g r o w t h r a t e s f o r a t y p i c a l e x p e r i ment a r e shown i n F i g u r e 2. The s a m p l e i n t e r v a l i s one m i n u t e , and t h e r e a r e 110 m e a s u r e m e n t s . I t c e r t a i n l y a p p e a r s t h a t b o t h the g r o w t h and r o t a t i o n s e r i e s a r e h i g h l y i r r e g u l a r i n t e r m s o f t h e i r growth r a t e s . I t i s d i f f i c u l t t o s e e any s y s t e m a t i c b e h a v i o r and t h e r e i s l i t t l e a p p a r e n t c o r r e l a t i o n b e t w e e n t h e two. These q u a l i t a t i v e o b s e r v a t i o n s must be q u a n t i f i e d . E s t i m a t e s o f t h e a u t o c o r r e l a t i o n f u n c t i o n and t h e p a r t i a l a u t o c o r r e l a t i o n s f o r the r o t a t i o n a l growth curve i n F i g u r e 2 a r e p r e s e n t e d i n F i g u r e s 3 and 4. The a u t o c o r r e l a t i o n f u n c t i o n o f t e n d e p i c t s d e p e n d e n c e o f a measurement v a l u e on r e c e n t p r e v i o u s v a l u e s w h e r e a s t h e p a r t i a l a u t o c o r r e l a t i o n s show t h e s i g n i f i c a n c e


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Figure 3.

A utocorrelation function of a rotational growth series.

LRC

Figure 4.

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Partial autocorrelation of a rotational growth series.

LPG

410

BIOCHEMICAL

ENGINEERING

of adding a d d i t i o n a l terms t o an a u t o r e g r e s s i v e model o f t h e series. I n F i g u r e 3, t h e r e i s l i t t l e s i g n i f i c a n c e and no r e a l p a t t e r n apparent. Two s t a n d a r d - e r r o r l i m i t s f o r t h i s s a m p l e s i z e a r e a b o u t ±0.2, and o n l y t h e a u t o c o r r e l a t i o n e s t i m a t e a t l a g 7 exceeds these l i m i t s . The same o b s e r v a t i o n s c a n be made o f F i g u r e 4. The i m p l i c a t i o n i s t h a t t h e v a r i a t i o n s i n r o t a t i o n a l g r o w t h a r e " w h i t e , " t h a t i s , random and u n c o r r e l a t e d ( w i t h e a r lier variations). To c o n f i r m t h i s , a c u m u l a t i v e p e r i o d o g r a m o f t h e f r e q u e n c y s p e c t r u m was computed and i s p r e s e n t e d i n F i g u r e 5. S i n c e t h e s o l i d c u r v e f o l l o w s t h e 45°-dashed-line, t h i s p r o v e s that the rotational series i s e s s e n t i a l l y white, quantifying the c o n j e c t u r e made i n a p r e v i o u s p a p e r ( 9 ) . The a u t o c o r r e l a t i o n f u n c t i o n and p a r t i a l a u t o c o r r e l a t i o n f o r t h e e l o n g a t i o n a l g r o w t h s e r i e s a r e p r e s e n t e d i n F i g u r e s 6 and 7. Systematic behavior i s evident i n the regular pattern of the a u t o c o r r e l a t i o n f u n c t i o n and t h e s i g n i f i c a n c e o f t h e p a r t i a l a u t o c o r r e l a t i o n s a t l a g s 1 and 2. T h i s d i r e c t l y s u g g e s t s t h a t a s e c o n d - o r d e r a u t o r e g r e s s i v e model i s a p p r o p r i a t e f o r t h i s s e r i e s (11). The m o d e l f o r m i s

g

k

+

a

l*k-l

+

a

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t

h

t

n

=

e

k

where g£ i s t h e i element o f t h e s e r i e s , a ^ , a£ a r e m o d e l p a r a m e t e r s , and e± i s t h e i model e r r o r o r r e s i d u a l . L e a s t s q u a r e s e s t i m a t i o n was a p p l i e d t o t h i s s e r i e s f o r t h e s e c o n d - o r d e r m o d e l a b o v e , and t h e f o l l o w i n g p a r a m e t e r v a l u e s w e r e determined: a

x

= 0.570

a

2

= 0.412

and

Such a m o d e l d e s c r i b e s a n underdamped o s c i l l a t o r y b e h a v i o r w i t h a s i n u s o i d a l p e r i o d o f 11 m i n u t e s and damping c o e f f i c i e n t o f 0.44. This p e r i o d o f o s c i l l a t i o n corresponds t o the "hunting" swings o f the sporangiophore; t h e r e f o r e , i t appears t h a t o u r model accounts f o r t h e g r o s s m e c h a n i c a l b e h a v i o r o f Phycomyces and i t s i n t e r a c t i o n w i t h o u r measurement s y s t e m . I t i s p o s s i b l e t o remove t h i s s y s t e m a t i c b e h a v i o r f r o m t h e d a t a and examine l o c a l v a r i a t i o n s i n e l o n g a t i o n a l growth r a t e by s t u d y i n g t h e r e s i d u a l s e r i e s o f t h e above model. T h i s r e s i d u a l s e r i e s was a n a l y z e d and f o u n d t o be e s s e n t i a l l y w h i t e : t h e cumu l a t i v e p e r i o d o g r a m i s p r e s e n t e d i n F i g u r e 8. T h i s i s i n c o n t r a s t to t h e p e r i o d o g r a m o f t h e o r i g i n a l e l o n g a t i o n a l s e r i e s , s e e F i g u r e 9, w h i c h shows s i g n i f i c a n t d e v i a t i o n f r o m t h e 45° l i n e . We c a n summarize t h e a n a l y s i s o f b o t h g r o w t h s e r i e s b y s t a t i n g t h a t l o c a l g r o w t h b e h a v i o r e x h i b i t e d b y t h e s p o r a n g i o p h o r e i s random


GAMOW

AND

CLOUGH

Growth

of

Sporangiophores

FREQUENCY

Figure 5.

Cumulative periodogram of a rotational growth series.



Figure 6.

Autocorrelation function of an elongational growth series.

Figure 7.

Partial autocorrelation of an elongational growth series.

S

w

9

to

19.

GAMOW

AND

Figure 8.

CLOUGH

Growth

of

Sporangiophores

Cumulative periodogram of an elongational residual series.


413

414

BIOCHEMICAL

ENGINEERING


19.

GAMOW

AND

CLOUGH

Growth

of

Sporangiophores

415

and u n c o r r e l a t e d when t h e e l o n g a t i o n a l r a t e and r o t a t i o n a l r a t e s e r i e s are studied separately. I t i s tempting to r e l a t e the n o i s y growth p a t t e r n s s i n c e s i m i l a r mechanisms may u n d e r l i e e l o n g a t i o n and r o t a t i o n . A p l o t of the c r o s s - c o r r e l a t i o n f u n c t i o n between the r o t a t i o n a l s e r i e s and t h e e l o n g a t i o n a l r e s i d u a l s e r i e s i s shown i n F i g u r e 10. The r e s u l t i s p e r h a p s u n e x p e c t e d , s i n c e t h e r e i s no s i g n i f i c a n t c o r r e l a t i o n b e t w e e n t h e two s e r i e s , t h a t i s , t h e n o i s y g r o w t h p a t t e r n s a p p e a r i n d e p e n d e n t . The l i m i t s shown on t h e f i g u r e a r e f o r one s t a n d a r d e r r o r , and one w o u l d e x p e c t a s i g n i f i c a n t c r o s s - c o r r e l a t i o n t o e x c e e d two s t a n d a r d e r r o r s . Conclusions We h a v e shown t h a t s t o c h a s t i c v a r i a t i o n s i n t h e r o t a t i o n a l and e l o n g a t i o n a l g r o w t h r a t e s o f Phycomyces a r e random and un correlated. S y s t e m a t i c b e h a v i o r i n e l o n g a t i o n a l g r o w t h was a t t r i b u t e d to the slower " h u n t i n g " motion of the sporangiophore. The i r r e g u l a r p a t t e r n s i n r o t a t i o n a l and e l o n g a t i o n a l g r o w t h do n o t a p p e a r t o be i n t e r d e p e n d e n t n o r r e l a t e d t o t h e same m a j o r source or stimulus. Discussion A l l o r g a n i s m s t h a t p o s s e s s a r i g i d e x o s k e l e t o n such as the a r t h r o p o d s , i n s e c t s and c r u s t a c e a n s must h a v e e v o l v e d one o f many p o s s i b l e s t r a t e g i e s i n order to increase i n s i z e , i . e . to uniform l y i n c r e a s e i n v o l u m e . The a r t h r o p o d s s i m p l y m o l t t h e i r o l d e x o s k e l e t o n and t h e n r e d e p o s i t a new one a f t e r a s h o r t b u r s t o f i n t e n s i v e growth. P l a n t s i n g e n e r a l , but f u n g i i n p a r t i c u l a r , have d e v i s e d a s l i g h t l y d i f f e r e n t s t r a t e g y t o e n a b l e them t o grow e v e n though they are a l s o encased i n a r a t h e r r i g i d c e l l w a l l . The new s y n t h e s i z e d c e l l w a l l , t e r m e d t h e p r i m a r y w a l l , a p p e a r s as an end r e s u l t o f g i a n t c a r b o h y d r a t e m i c r o f i b r i l s b e i n g d e p o s i t e d on the i n n e r s u r f a c e of the e x i s t i n g p r i m a r y c e l l w a l l . In a d d i t i o n , i t has b e e n shown t h a t t h e l o n g i t u d i n a l a x i s o f t h e s e n e w l y l a i d down p o l y m e r s i s p e r p e n d i c u l a r t o t h e n e t d i r e c t i o n o f growth of the c e l l w a l l i t s e l f . Cell wall thickening resulting from newly deposited m i c r o f i b r i l s i s c o n s t a n t l y b e i n g counter b a l a n c e d by c e l l w a l l e x t e n s i o n . The d r i v i n g mechanism o f c e l l w a l l e x t e n s i o n i s the l a r g e pressure d i f f e r e n c e , the c e l l ' s t u r g o r p r e s s u r e , b e t w e e n t h e i n s i d e o f t h e c e l l w a l l and i t s e x t e r n a l e n v i r o n m e n t . B e c a u s e t h e p o l y m e r s a r e l a i d down p e r p e n d i c u l a r l y to i t s net d i r e c t i o n of growth, during c e l l w a l l extent i o n they are p a s s i v e l y r e o r i e n t e d towards the c e l l ' s l o n g i t u d i n a l a x e s , and i n a d d i t i o n , t r a n s p o r t e d t o w a r d t h e o u t e r s u r f a c e o f t h e c e l l w a l l . The m o d e l d e s c r i b e d a b o v e i s b a s i c a l l y known as t h e m u l t i n e t t h e o r y o f c e l l w a l l g r o w t h f i r s t p r o p o s e d by R o e l o f s e n and Houwink (12) and e x p e r i m e n t a l l y v e r i f i e d i n N i t e l l a by G e r t e l and G r e e n ( 1 3 ) .



-.5

.5

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Figure 10.

-3 -2

r(k)

1

2

3

4

5

6

7

8

9

18

11

r(k)

12 13

14 15

16 17

Cross-correlation between local variations in rotational and elongational growth.

θ

CROSSCORRELflTION FUNCTION

18.

GAMOW

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S i n c e t h e s e n e w l y l a i d down m i c r o f i b r i l s a r e i n c o n s t a n t m o t i o n , t h e p r i m a r y c e l l w a l l b e h a v e s much l i k e a v i s c o - e l a s t i c fluid. B a r t n i c k i - G a r c i a (14) i n 1970 p r o p o s e d t h a t t h e n e w l y s y n t h e s i z e d c e l l w a l l i s i n a constant s t a t e of f l u x i n terms of s y n t h e s i s and d e g r a d a t i o n r e s u l t i n g f r o m t h e p r e s e n c e o f s y n t h e s i z i n g and l y t i c enzymes, r e s p e c t i v e l y . I n 1974 O r t e g a and Gamow (15) p r o p o s e d t h a t t h e o r i e n t a t i o n o f m i c r o f i b r i l s , o r b u n d l e s o f m i c r o f i b r i l s , c o u l d a c c o u n t f o r b o t h c e l l e l o n g a t i o n and c e l l rotation. They a l s o d e d u c e d t h a t t h e m a g n i t u d e o f t h e r a t i o o f r o t a t i o n t o e l o n g a t i o n must be a f u n c t i o n o f t h e m i c r o f i b r i l a n g l e , the a n g l e b e i n g measured i n r e s p e c t to the l o n g i t u d i n a l a x i s of the growing c e l l w a l l . T h i s model p r e d i c t e d t h a t the f i b r i l s l o c a t e d i n the lower r e g i o n of the growing zone, those which would have a l r e a d y m a x i m a l l y r e o r i e n t a t e d towards the l o n g i t u d i n a l a x i s , w o u l d show maximum r o t a t i o n t o e l o n g a t i o n r a t i o s , and t h i s p r e d i c t i o n was e x p e r i m e n t a l l y v e r i f i e d . Our p r e s e n t r e s u l t s c o n t r a d i c t t h e s i m p l e r e o r i e n t a t i o n t h e o r y as p r e s e n t e d a b o v e s i n c e i t c l e a r l y p r e d i c t s t h a t any i r r e g u l a r i t y i n t h e g r o w t h r a t e w o u l d be d i r e c t l y c o u p l e d w i t h an i r r e g u l a r i t y i n t h e r o t a t i o n r a t e . Our p r e s e n t d a t a w o u l d be c o n s i s t e n t w i t h a r e o r i e n t a t i o n m o d e l i n w h i c h t h e maximum r o t a t i o n and e l o n g a t i o n o c c u r i n d i f f e r e n t r e g i o n s o f t h e GZ. Recent f i n e s t r u c t u r e s t u d i e s i n our l a b o r a t o r y have c o n f i r m e d the 1958 r e p o r t by Cohen and D e l b r u c k t h a t t h e maximun e l o n g a t i o n r a t e o c c u r r i n g i n the r e g i o n of the growing zone i s d i s t i n c t l y d i f f e r e n t f r o m t h a t o c c u r r i n g i n t h e r e g i o n o f maximum r o t a t i o n r a t e . Our p r e s e n t a n a l y s i s i s c o n s i s t e n t w i t h t h i s f i n d i n g b e c a u s e i f b o t h r o t a t i o n and e l o n g a t i o n a r e s t o c h a s t i c i n t e r m s o f g r o w t h i r r e g u l a r i t i e s , we w o u l d n o t e x p e c t t o f i n d a c o r r e l a t i o n i f t h e r o t a t i o n and e l o n g a t i o n o c c u r r e d i n d i f f e r e n t and i n d e p e n d e n t r e g i o n s of the growing zone. L a s t l y , we a r e i n t r i g u e d by t h e s i n u s o i d a l b e h a v i o r , h a v i n g a p e r i o d o f 11 m i n u t e s , i n t h e e l o n g a t i o n r a t e . A t t h i s t i m e we h a v e no way o f d e t e r m i n i n g w h e t h e r t h i s o s c i l l a t i o n i s r e a l i n the sense t h a t the net r a t e of growth o s c i l l a t e s w i t h t h i s p e r i o d o r w h e t h e r we a r e o b s e r v i n g t h e w e l l - k n o w n h u n t i n g b e h a v i o r ( 1 0 ) ; Phycomyces s p o r a n g i o p h o r e s show b o t h a f a s t , 5 t o 7.5 m i n , and a s l o w , 30 t o 60 m i n , o s c i l l a t i o n . Our g u e s s w o u l d be t h a t t h e 1 1 - m i n u t e p e r i o d we h a v e o b s e r v e d i s a d i r e c t r e s u l t o f t h e mea s u r e m e n t e r r o r c a u s e d by t h e s p o r a n g i o p h o r e s h u n t i n g behavior. I n o r d e r t o c o n f i r m t h i s s p e c u l a t i o n , we w o u l d r e q u i r e f a s t e r s a m p l i n g r a t e s f o r b o t h e l o n g a t i o n and r o t a t i o n a l g r o w t h . Such an i n c r e a s e i n s a m p l i n g r a t e w o u l d a l s o f a c i l i t a t e more d e t a i l e d study of the s t o c h a s t i c growth b e h a v i o r . We b e l i e v e an e x t e n s i o n o f o u r p r e s e n t measurement t e c h n i q u e w o u l d a l l o w f o r t h i s . This w o u l d be t o c o u p l e a c o m p u t e r - b a s e d p a t t e r n r e c o g n i t i o n s y s t e m t o o u r m i c r o s c o p i c v i d e o e q u i p m e n t . By h a v i n g t h e computer f o l l o w motion of s e v e r a l t r a c k i n g p a r t i c l e s , a high r e s o l u t i o n p r o f i l e o f b o t h g r o w t h r a t e s i s f e a s i b l e . Such an e x t e n s i o n t o o u r p r e s e n t a p p a r a t u s w o u l d r e q u i r e l e s s t h a n $100,000 i n c a p i t a l f u n d i n g . 1


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A p p e n d i x : A B r i e f D e s c r i p t i o n o f Time S e r i e s A n a l y s i s The t e c h n i q u e s o f t i m e s e r i e s a n a l y s i s h a v e b e e n a p p l i e d a c r o s s a b r o a d s p e c t r u m o f f i e l d s , n o t a b l y b u s i n e s s and e c o n o m i c s , e n g i n e e r i n g , and m e t e o r o l o g y . They h a v e g a i n e d w i d e a c c e p t a n c e p a r t l y due t o t h e h i g h y i e l d o f m o d e l i n g i n f o r m a t i o n one a c h i e v e s t h r o u g h t h e a p p l i c a t i o n o f a s t r a i g h t - f o r w a r d , e a s y - t o - u s e method ology. T h e r e a r e two common s c e n a r i o s : 1) a n a l y s i s o f a s i n g l e t i m e s e r i e s , u s u a l l y a s a m p l e d mea s u r e m e n t s i g n a l , t o d e t e r m i n e i t s s y s t e m a t i c and random character 2) a n a l y s i s o f two o r more t i m e s e r i e s t o a s s e s s and q u a n t i f y c o n n e c t i v e r e l a t i o n s h i p s (one must be c a r e f u l w i t h claims of "cause-and-effect" h e r e — a d d i t i o n a l informa t i o n on s t r u c t u r e i s u s u a l l y r e q u i r e d ) I t i s d e s i r a b l e , from a p r a c t i c a l s t a n d p o i n t , t o model time s e r i e s i n as s i m p l e a form as p o s s i b l e . Such a r e l i n e a r models w i t h a m i n i m a l number o f p a r a m e t e r s . T h e r e a r e two p r i m a r y s t e p s t o t h e m o d e l i n g p r o c e s s : i d e n t i f i c a t i o n o f t h e m o d e l f o r m and e s t i m a t i o n of the model's parameters. These s t e p s a r e a p p r o p r i a t e l y f o l l o w e d by t e s t i n g o f t h e m o d e l ' s a b i l i t y t o f i t o r p r e d i c t new d a t a . I t i s t y p i c a l o f a s i n g l e t i m e s e r i e s t h a t measurements w i l l show some dependence on t h e i r r e c e n t p a s t h i s t o r y — m a n y p r o c e s s e s are i n e r t i a l i n behavior. The a u t o c o r r e l a t i o n f u n c t i o n ( o r i t s e s t i m a t e f r o m t h e s e r i e s ' d a t a ) w i l l r e v e a l s u c h d e p e n d e n c e , and t h e p a r t i a l a u t o c o r r e l a t i o n s w i l l h e l p show how f a r i n t o t h e p a s t t h i s " a u t o r e g r e s s i v e " m o d e l must e x t e n d . The m o d e l t h u s s e l e c t e d c a n be f i t t o t h e d a t a by one o f many r e g r e s s i o n t e c h n i q u e s , e.g. l i n e a r l e a s t squares. The r e s i d u a l s , t h o s e components o f t h e d a t a w h i c h a r e n o t e x p l a i n e d by t h e a u t o r e g r e s s i v e m o d e l , f o r m another s e r i e s . I f the r e s i d u a l s e r i e s i s u n c o r r e l a t e d ( t o i t s e l f ) , t h i s i n d i c a t e s a random b e h a v i o r w h i c h we c a n n e i t h e r p r e d i c t n o r r e g u l a t e — w e j u s t have t o l i v e w i t h i t as u n c e r t a i n be havior. O f t e n , however, i t i s p o s s i b l e t o model t h e r e s i d u a l s a l s o i n a s i m i l a r , a u t o r e g r e s s i v e manner—which would i n d i c a t e t h a t a l t h o u g h we c a n n o t r e g u l a t e t h e s e q u a n t i t i e s , we c a n do some prediction. This s t r u c t u r e o f the r e s i d u a l s e r i e s i s u s u a l l y b u i l t b a c k i n t o t h e o r i g i n a l m o d e l , and t h e t e r m i n o l o g y u s e d f o r t h e r e s i d u a l d e s c r i p t i o n i s "moving a v e r a g e . " I t i s then best t o r e f i t t h e e n t i r e m o d e l by r e g r e s s i o n . A g e n e r a l f o r m f o r t h e a u t o r e g r e s s i v e - m o v i n g average model i s g. + a-g. -+...+ a g. = e. + c-e. - +...+ k l k-l n k-n k 1 k-1 &

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where e^ i s now an u n c o r r e l a t e d r e s i d u a l . The a b o v e m o d e l may a p p e a r i m p o s i n g ; h o w e v e r , t h e a p p r o p r i a t e v a l u e s f o r η and ρ a r e u s u a l l y l e s s than t h r e e . C o n n e c t i o n b e t w e e n two s e r i e s i s m a n i f e s t e d i n t h e c r o s s c o r r e l a t i o n f u n c t i o n ( o r i t s e s t i m a t e ) . When a c a u s e - a n d - e f f e c t ,


19.

Growth of

GAMOW A N DCLOUGH

Sporangiophores

419

" i n p u t - o u t p u t " r e l a t i o n s h i p i s j u s t i f i e d , a model o f t h e form g, + a- g, -+...+ a g, = b χ, , + b-x. ,-+...+ b χ. , k l k-l n k-n ο k-d 1 k-d-1 m k-d-m 6

ô

&

+ e. + c-e. ,+...+ c e. k 1 k-1 ρ k-p may b e p r o p o s e d w h i c h i n c l u d e s t h e a u t o r e g r e s s i v e d e p e n d e n c e o f the o u t p u t v a r i a b l e ( g ^ ) , a " c a u s e " dependence o n v a l u e s o f t h e i n p u t v a r i a b l e (x^) and a d e s c r i p t i o n o f r e s i d u a l (e^) b e h a v i o r as b e f o r e . S e l e c t i o n o f m o d e l f o r m ( w h i c h may i n c l u d e a d e l a y o f d sample i n t e r v a l s b e t w e e n t h e i n p u t a n d o u t p u t v a r i a b l e s ) i s f a c i l i t a t e d b y s t u d y o f t h e c r o s s - c o r r e l a t i o n f u n c t i o n and i m p u l s e response f u n c t i o n estimates i n a d d i t i o n t o the a u t o c o r r e l a t i o n f u n c t i o n and p a r t i a l a u t o c o r r e l a t i o n s . Once t h e m o d e l f o r m i s s e l e c t e d , r e g r e s s i o n t e c h n i q u e s a r e used t o f i t t h e parameters. Two m o d e l - c h e c k i n g methods a r e common, and t h e y a r e o f t e n a p p l i e d t o a "new" s e t o f d a t a . F i r s t , t h e r e s i d u a l s a r e a n a l y z e d f o r l a c k o f c o r r e l a t i o n o r " w h i t e n e s s . " White n o i s e has the c h a r a c t e r i s t i c t h a t a l l f r e q u e n c i e s c o n t a i n e d i n i t s h o u l d be present i n equal strength. The c u m u l a t i v e p e r i o d o g r a m i s t h e i n t e g r a l o f a spectrum e s t i m a t e ; t h e r e f o r e , f o r w h i t e n o i s e , i t s h o u l d b e t h e i n t e g r a l o f a c o n s t a n t o r a l i n e a r ramp f u n c t i o n . The " w h i t e n e s s " o f t h e r e s i d u a l s c a n b e a n a l y z e d b y p l o t t i n g t h e i r c u m u l a t i v e p e r i o d o g r a m and o b s e r v i n g c l o s e n e s s t o t h e i d e a l ramp l i n e . A n o t h e r common d i a g n o s t i c t e c h n i q u e i s t o d e t e r m i n e the s t a t i s t i c a l e f f i c i e n c y o f a d d i n g one o r two p a r a m e t e r s ( a n d terms) t o t h e model. T h i s w i l l check i f t h e minimal p a r a m e t r i c m o d e l c h o s e n i s t r u l y p a r s i m o n i o u s . The s t a t i s t i c a l F - t e s t b a s e d on t h e e x t r a - s u m - o f - s q u a r e s p r i n c i p l e i s u s e f u l f o r t h i s e f f i ciency test.

Literature 1.

2. 3. 4. 5. 6. 7. 8. 9.

Cited

Bergman, K.; Burke, P. V.; Cerda-Olmedo, E.; David, C. N.; Delbrück, M.; F o s t e r , K. W.; Goodell, E. W.; Heisenberg, M.; Meissner, G.; Zalokar, M.; Dennison, D. S.; Shropshire, W.; Jr. B a c t e r i a l Rev. 1969, 33, 97-157. Shropshire, W., J r . Phys. Rev. 1963, 43, 38-67. Cerda-Olmedo, E. Ann. Rev. M i c r o b i o l . 1977, 31, 535-47. Foster, K. W. Ann. Rev. Biophys. Bioeng. 1977, 6, 419-43. Cohen, R. J . ; F r i e d , M. G.; Atkinson, M. M. Biochem. and Biophys. Res. Comm. 1979, 86, 877-84. Gamow, R. I . ; Böttger, B. J . Gen. P h y s i o l . , i n p r e s s . Cohen, R.; Delbruck, M. J . Cell Comp. P h y s i o l . 1958, 52, 361-88. Ortega, J . Κ. E.; H a r r i s , J . ; Gamow, R. I . Plant P h y s i o l . 1974, 53, 485-90. Gamow, R. I . ; Böttger, B. J . Gen. P h y s i o l . 1981, 77, 65-75.


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10. 11.

12. 13. 14. 15.

BIOCHEMICAL

ENGINEERING

Dennison, D. S. Science 1959, 129, 775-7. Box, G. E. P.; Jenkins, M. "Time S e r i e s A n a l y s i s : F o r e c a s t ing and C o n t r o l " ; Revised E d i t i o n ; Holden-Day: San Francisco, CA, 1976. Roelofsen, P. Α.; Houwink, A. L. Acta. Bot. N e e r l . 1953, 2, 218-25. G e r t e l , E. T.; Green, P. B. P l a n t P h y s i o l . 1977, 60, 247-54. B a r t n i c k i - B a r c i a , S. In "Phytochemical Phylogeny;" Hasborne, J . B., Ed.; Academic Press, Inc.; New York, NY, 1970; p. 81. Ortega, J . Κ. E.; Gamow, R. I. J . Theoret. B i o l . 1974, 47, 317-32.

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