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Equilibrium ultracentrifugation as a method for studying protein interactions Van de Voort, Frederik Robert 1977

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EQUILIBRIUM ULTRACENTRIFUGATION FOR STUDYING  PROTEIN  AS A METHOD  INTERACTIONS  by Fred j v ^ B.Sc.(Agr.), M.Sc,  University University  n  ^  e  Voort  of B r i t i s h of B r i t i s h  Columbia, Columbia,  1972 1974  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE  STUDIES  DEPARTMENT OF FOOD SCIENCE UNIVERSITY OF BRITISH COLUMBIA  We  a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required standard  THE UNIVERSITY OF B R I T I S H COLUMBIA August,  1977  F r e d v a n de V o o r t ,  1977  In p r e s e n t i n g an  this  thesis  in partial  advanced degree a t the U n i v e r s i t y  the  Library  I further for  shall  make i t f r e e l y  agree t h a t p e r m i s s i o n  scholarly  h i s representatives.  of  this  written  permission.  f^A  c  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  gain  SeoT  i  l«n 7  ^  ^  ^  Columbia  Columbia,  I agree  that  f o r r e f e r e n c e and s t u d y .  for extensive  copying o f this  thesis  by t h e Head o f my D e p a r t m e n t o r  I t i s understood  thesis f o r financial  Department o f  of B r i t i s h  available  p u r p o s e s may be g r a n t e d  by  f u l f i l m e n t o f the requirements f o r  shall  that  copying or p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t my  ABSTRACT  The  ultracentrifuge  has t r a d i t i o n a l l y  to o b t a i n molecular weights,  t h e s e d e v e l o p m e n t s was  complete  molecular weight  ultracentrifugal programming  systems.  procedure  instrument.  acquisition  to obtain the (MWD)  t h e use o f a  from linear  by S c h o l t e .  r e s e a r c h was t o a p p l y t h e  centrifugal  w i t h new UV o p t i c s system  and i n t e g r a t e d w i t h a d a t a  coupled with a desktop o f molecular weights  d a t a o f b o t h homogeneous and  The c o m b i n a t i o n  acquisition  system  was  computer t o a l l o w from  of ultracentrifuge  found  t o work w e l l  A new c a l c u l a t i o n investigated.  molecular weight than  providing  developed  and d a t a  i n spite of system.  f o r obtaining the  MWD  Development o f the f r e q u e n c y v s .  f u n c t i o n was b a s e d  linear  on m u l t i p l e r e g r e s s i o n  programming and h a d t h e a d v a n t a g e o f  statistical  of the f i t t i n g  procedure  ultra-  heterogeneous  b a s e l i n e problems a s s o c i a t e d w i t h the o p t i c a l  rather  MWD  of Scholte t o the study o f p r o t e i n  the routine c a l c u l a t i o n  was a l s o  One  I n p r e p a r a t i o n f o r t h i s work, t h e u l t r a c e n t r i f u g e  was e q u i p p e d  systems.  distribution  developed  One o b j e c t i v e o f t h i s calculation  of this  the a b i l i t y  data through  calculation  used  b u t r e c e n t developments  have e x p a n d e d t h e c a p a b i l i t i e s of  been  parameters  procedures.  and t e s t e d  to assess the accuracy  A FORTRAN p r o g r a m  was  i n regard to i t sc a p a b i l i t i e s  and  limitations,  t h r o u g h the a n a l y s i s o f model systems.  multiple regression  a p p r o a c h was  t o t h e method o f S c h o l t e . statistical  f o u n d t o be  The a v a i l a b i l i t y  the  evaluation  and  the concentrations  search  allowed required of  Using  i n t h e number  to obtain  a distribution,  to verify new  of rotor  allowed weights present  routine  speeds  t h u s e l i m i n a t i n g one  experimentally approach only  t h a t t h e method was v i a b l e .  MWD  theoretically  of Scholte's  method. the t h e o r e t i c a l  provided  Due  t h e a v a i l a b l e e q u i p m e n t , t h e MWD  n o t be p u t t o use i n s t u d y i n g the  that  the simplex  f i t solutions  a reduction  advantages o f t h i s  of  systems.  f o r the best  Attempts  routine  o f i n d i v i d u a l components  the major r e s t r i c t i o n s  tion  of the  of the weight average molecular  i n multicomponent to  equivalent  parameters l e d t o the i n c l u s i o n i n the  program o f a simplex o p t i m i z a t i o n  The  an i n d i c a -  to the l i m i t a t i o n s  c a l c u l a t i o n could  p r o t e i n systems,  p o t e n t i a l f o r i t s u s e i n s u c h work was  although  demonstrated.  i.c  TABLE OF CONTENTS  Page L I S T OF TABLES  v  L I S T OF FIGURES  v i  ACKNOWLEDGEMENTS  X  INTRODUCTION  .  LITERATURE REVIEW PART I .  .  1  .  4  MOLECULAR WEIGHT DETERMINATIONS  THEORY  12 ..  12  EXPERIMENTAL  18  Ultracentrifugation  18  Data P r o c e s s i n g  24  A.  Hardware  25  1.  Data a c q u i s i t i o n  2.  Desktop  system  25.  computation and  plotting 3.  B.  Application acquisition centrifuge  28 o f the data system t o the •-• •  Software  31  1.  Data c o n v e r s i o n program  2.  Automatic l i n e a r regression p l o t t i n g program  3.  31 and  Automatic multiple regression p r o g r a m w i t h back, c a l c u l a t i o n feature  RESULTS Homogeneous Systems  30  ..  32  32 3  3  33  ii. Page Heterogeneous Systems  40  DISCUSSION PART I I .  49 MOLECULAR WEIGHT DISTRIBUTIONS  THEORY  53 53  L i n e a r Programming - The S o l u t i o n o f S c h o l t e ...  58  M u l t i p l e Regression as an A l t e r n a t i v e Solution  62  RESULTS AND DISCUSSION  68  Model Systems  68  A.  Log normal d i s t r i b u t i o n method  69  B.  Simple expansion o f the Rinde equation M o l e c u l a r Weight D i s t r i b u t i o n Program Using M u l t i p l e Regression  77  Comparison o f L i n e a r Programming and M u l t i p l e Regression  91  71  F a c t o r s A f f e c t i n g the M o l e c u l a r Weight D i s t r i b u t i o n  93  A.  Speed  94  B.  I n t e r v a l and range  97  C.  Loss o f data  100  D.  Summary  100  Expanding the M u l t i p l e R e g r e s s i o n Approach A. B. C.  A s s e s s i n g the s t a t i s t i c a l parameters The concept o f an i t e r a t i v e solution The i n i t i a l  i t e r a t i v e algorithm  101 101 102 106  iii. Page D.  E.  The simplex i t e r a t i v e a l g o r i t h m  10 8  1.  The simplex method  109  2.  The s t a r t i n g m a t r i x  114  3.  The simplex output  118  Smoothing o f undefined  data  123  Case S t u d i e s o f Some Model Systems  126  A.  Catalase  126  B.  Trypsin inhibitor-ovalbuminconalbumin  130  a s ^ - c a s e i n and K - c a s e i n and i n t e r a c t i o n product  132  C. D.  Lysozyme-ovalbumiri and interaction  E.  product  '  Discussion  134 .  135  A n a l y s i s of P r o t e i n Mixtures  136  A.  Trypsin inhibitor-conalbumin  138  B. C.  Trypsin inhibitor-ovalbuminconalbumin Ovalbumin-thyroglobulin  139 I  D.  Catalase  1  E.  Discussion  I  4  Other F a c t o r s  4  4  4  Extinction  B.  P a r t i a l s p e c i f i c volume  LITERATURE CITED  2  ^- ^  A.  CONCLUSION  4  2  coefficients  l  ^  4  I -  4  7  1  5  0  1  5  7  iv. Page APPENDIX  164  A.  Proteins  B.  Assembly o f Yphantis C e l l  164  C.  Programs f o r the Monroe C a l c u l a t o r  165  1.  Used i n T h i s I n v e s t i g a t i o n  Time t o reach e q u i l i b r i u m program  2.  Orthogonal p o l y n o m i a l curve program  3.  Data c o n v e r s i o n program f o r manually  D.  165 fitting  s e l e c t e d data  FORTRAN Programs 1.  The m o l e c u l a r weight  E. F. G.  168  169 169  distribution  program  2.  164  173  (a)  C a l c u l a t i o n mode  173  (b)  I t e r a t i v e mode  174  (c)  Smoothing mode  , 175  The simplex program  Listing of L i s t i n g of Scientific f o r the MWD Listing of  the MWD Program the P o r t i o n o f the Subroutine Package Used Program the Simplex Program  176 177 18$ 201  L I S T OF TABLES  Table  Page  I.  M o l e c u l a r Weights Obtained  f o r Ovalbumin  42  II.  M o l e c u l a r Weights Obtained  for Catalase  50  III.  IV.  V.  Comparison o f F r e q u e n c i e s R e s u l t i n g from A n a l y s i s o f a Two Component S y s t e m U s i n g M u l t i p l e R e g r e s s i o n and L i n e a r Programming  ....  92  F r e q u e n c i e s and F - R a t i o s O b t a i n e d b y M u l t i p l e Regression Analysis of a 20,000-80,000 Dalton Mixture (1:1 Ratio) Using S e l e c t e d Molecular Weight P a i r s -  .105  Summary o f t h e S i m p l e x C o n t r a c t i o n O p e r a t i o n s . .  115  VI  LIST OF FIGURES  Figure  Page  1.  Schematic diagram o f the UV o p t i c a l system . . . .  20  2.  Recorder t r a c e (scan) of a t y p i c a l sedimentation e q u i l i b r i u m experiment u s i n g a double s e c t o r c e l l . . . .  21  Photograph of the d a t a a c q u i s i t i o n system on a l a b o r a t o r y c a r t - t e l e t y p e , v o l t m e t e r and d i g i t i z e r  26  Photograph of the a c q u i s i t i o n system, m u l t i p l e x i n g u n i t and u l t r a c e n t r i f u g e  27  Schematic diagram or the data p r o c e s s i n g operations  29  3.  4. 5. 6.  Schematic diagram of data flow through the data c o n v e r s i o n , l i n e a r r e g r e s s i o n and m u l t i p l e r e g r e s s i o n programs  7.  F a c s i m i l e of t e l e t y p e output  8.  Input and output o f the d a t a c o n v e r s i o n program and l i n e a r r e g r e s s i o n p l u s p l o t t i n g program 2 P l o t o f In A v s . x f o r ovalbumin showing the l e a s t squares f i t t o the data 2 P l o t of In A v s . x f o r c a t a l a s e showing the l e a s t squares q u a d r a t i c f i t t o the data  9. 10. 11. 12. 13. 14.  . . .  34 36  38  41 . . .  44  Input and output of the m u l t i p l e r e g r e s s i o n program . . . . . . . . . Input and output o f the m u l t i p l e back c a l c u l a t i o n  45  regression  Input and output o f the l o g normal d i s t r i b u t i o n program . . S e m i l o g a r i t h m i c p l o t o f the output, f(M) v s . M, from the l o g normal d i s t r i b u t i o n program f o r a three component system o f 25,000-80,000-320,000 d a l t o n s (1:1:1 r a t i o ) .  „  47  .. .  72  . . .  74  vii. Figure 15.  16. 17. 18. 19.  20.  21.  22.  23.  24.  25.  Page S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 25,000-80,000 d a l t o n mixture (1:1 r a t i o ) . M u l t i s p e e d data; i n t e r v a l =2.0  74  Input and output o f the £ v s . c(£)/c program f o r the l o g normal d i s t r i b u t i o n  75  Input and output o f the Rinde equation program  78  Example of the output o f the m u l t i p l e r e g r e s s i o n MWD program Semilogarithmic p l o t of f(M) v s . M f o r a 25,000-80,000 d a l t o n mixture (1:1 r a t i o ) . S i n g l e speed data; i n t e r v a l = 2.0 . . . . . . .  . 8 4  .  96  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 25,000-80,000-320,000 d a l t o n mixture (1:1:1 r a t i o ) . S i n g l e speed d a t a ; i n t e r v a l = 2.0  96  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 25,000-80,000 d a l t o n mixture (1:1 r a t i o ) . M u l t i s p e e d data; i n t e r v a l =1.5  98  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 67,000-134,000 d a l t o n mixture (2:1 r a t i o ) . M u l t i s p e e d data; i n t e r v a l =2.0 . . . .  98  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 67,000-134,000 d a l t o n mixture (2:1 r a t i o ) . M u l t i s p e e d data; i n t e r v a l =1.4  99  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 67,000-134,000 d a l t o n mixture (2:1 r a t i o ) . M u l t i s p e e d data; i n t e r v a l = 1.4, w i t h the range having been a d j u s t e d  99  I l l u s t r a t i o n o f the f i r s t b a s i c o p e r a t i o n s of the simplex o p t i m i z a t i o n r o u t i n e  26.  Flowchart  f o r the simplex a l g o r i t h m  27. 28.  I n i t i a l output of the simplex a l g o r i t h m Semilogarithmic p l o t o f f(M) v s . M f o r a 25,000-80,000 d a l t o n mixture (1:1 r a t i o ) . S i n g l e speed data; i n t e r v a l = 2.0, w i t h d a t a having undergone smoothing  I l l 119 120  125  v x n ,  Figure 29.  30.  31.  32.  33.  34.  35.  36.  37.  38.  39.  Page S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 25,000-8,000-320,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . S i n g l e speed d a t a ; i n t e r v a l = 2.0, w i t h d a t a h a v i n g u n d e r g o n e s m o o t h i n g  . . .  125  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 57,000-232,000 d a l t o n m i x t u r e (1:2 r a t i o ) . 5,000, 10,000 a n d 15,000 rpm, i n t e r v a l = 2 . 0  . .  126  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 57,000-232,000 d a l t o n m i x t u r e (1:2 r a t i o ) . 5,000, 10,000 a n d 15,000 rpm, i n t e r v a l =2.0, with data having undergone smoothing . . . . . .  126  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 57,000-232,000 d a l t o n m i x t u r e (1:2 r a t i o ) . 1 3 , 0 0 0 r p m , i n t e r v a l = 2.0  131  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 17,000-45,000-85,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 10,000, 20,000 a n d 30,000 i n t e r v a l = 2.0  rpm, 131  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 17,000-45,000-85,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 10,000, 20,000 and 30,000.rpm, i n t e r v a l = 1.5 . . .  133  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 17,000-45,000-85,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 25,000 rpm, i n t e r v a l = 1.5, with data having undergone smoothing  133  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 25,000-400,000-1,500,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 3,000, 7,000 a n d 1 3 , 0 0 0 rpm, i n t e r v a l = 2.2  137  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a 25,000-400,000-1,500,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 1 0 , 0 0 0 r p m , i n t e r v a l = 2.2 S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a m i x t u r e o f t r y p s i n i n h i b i t o r and conalbumin. A b s o r b a n c e = 0.5 f o r b o t h , m i x e d 1:1 b y volume. Run a t 9,600, 15,500 a n d 21,600 rpm  . . .  . .  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a mixture o f t r y p s i n i n h i b i t o r and conalbumin. A b s o r b a n c e = 0.5 f o r b o t h , m i x e d 1:1 b y v o l u m e . Run a t 9,600, 15,500 a n d 21,600 r p m , w i t h d a t a having undergone smoothing  137  140  140  ix. Figure 40.  41.  42.  43.  Page S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a mixture of t r y p s i n i n h i b i t o r and conalbumin. Absorbance = 0.5 f o r both, mixed 1:1 by volume. Run a t 19,600 rpm  141  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a mixture o f t r y p s i n i n h i b i t o r , ovalbumin and conalbumin. Absorbance = 0.5 f o r a l l , mixed 1:1:1 by volume. Run a t 9,600, 15,500 and 21,600 rpm  141  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r a mixture o f ovalbumin and t h y r o g l o b u l i n . Absorbance = 0.5 f o r both, mixed 1:1 by volume. Run a t 6,300, 9,600 and 15,500 rpm  . .  143  S e m i l o g a r i t h m i c p l o t o f f(M) v s . M f o r c a t a l a s e . Absorbance = 0.5, run a t 9,600, 15,500 and 21,600 rpm  14 3  IA.  Time t o reach e q u i l i b r i u m program  166  2A.  Input and output of the data c o n v e r s i o n program f o r manually s e l e c t e d d a t a  170  I l l u s t r a t i o n o f card deck f o r v a r i o u s modes o f the MWD program and f o r the simplex program  172  3A.  ACKNOWLEDGMENT  I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e  t o Dr.  Shuryo N a k a i f o r h i s a i d and encouragement t h r o u g h o u t the  c o u r s e o f t h i s work.  true Dr.  learning M. Tung  experience.  Working w i t h him has been a I a l s o would  like  text,  f o r w h i c h I am most  Leonard  over the past three years S. K l i n c k  Finally, for  I would  also  a s s i s t a n c e w h i c h I, i n t h e form o f a  Fellowship.  I would  like  t o t h a n k my w i f e ,  h e r f o r b e a r a n c e and h e l p  which I d e d i c a t e  dollars.  editing the  grateful.  t o acknowledge t h e f i n a n c i a l  received  t o thank  f o r h i s a d v i c e and l o a n o f computer  My c o m m i t t e e members were most h e l p f u l final  like  to her.  i n preparing  Colleen,  this  manuscript  INTRODUCTION  The  o v e r a l l s t r u c t u r e and p r o p e r t i e s o f many food  products a r e dependent on the i n t e r a c t i o n s o f t h r e e major components:  p r o t e i n s , l i p i d s and p o l y s a c c h a r i d e s ,  as aqueous d i s p e r s i o n s . has  S c i e n t i f i c study o f food  l e d t o the r e a l i z a t i o n t h a t fundamental  components can be i s o l a t e d by p r o c e s s i n g materials  usually systems  food  v a r i o u s raw  and l a t e r recombined i n t o new food  products.  T h i s approach t o food manufacture r e q u i r e s an expanded base o f fundamental knowledge t h a t can be a p p l i e d t o the problems i n v o l v e d i n food f a b r i c a t i o n . In the case o f p r o t e i n s , t h e r e a r e numerous s i t u a t i o n s i n which t h e i r i n t e r a c t i o n s a r e important t o a food A c l a s s i c example i s the t h i n n i n g o f egg white storage,  system.  during  which i s b e l i e v e d t o be caused by the i n t e r -  a c t i o n of lysozyme and ovomucin.  Other examples which  i l l u s t r a t e the p r a c t i c a l s i g n i f i c a n c e o f p r o t e i n - p r o t e i n i n t e r a c t i o n s i n c l u d e the i n t e r a c t i o n o f k - c a s e i n w i t h 6 l a c t o g l o b u l i n r e l a t i v e t o the heat s t a b i l i t y o f m i l k , and  the i n t e r a c t i o n o f p r o t e o l y t i c enzymes w i t h  substrates.  their  Since these i n t e r a c t i o n s a r e an i n t e g r a l  p a r t o f foods, forces involved  the e l u c i d a t i o n o f the mechanisms and (such as hydrophobic and e l e c t r o s t a t i c )  are fundamental t o the understanding o f food  systems.  2.  However, t h e that  complexity  initially  of a food product  a more m a n a g e a b l e s y s t e m be  P r e v i o u s work i n t h i s  laboratory  and a  as g e l f i l t r a t i o n ,  limitation  first  two  in i t s application  methods, t h e  was  had  to these  the  studies.  thermodynamic d a t a ,  In  hindered  and  in  only limited  the  the  information  provided.  another  firmly  b a s e d on  only a limited average or  the use  and  (45).  1  found  t o the Rinde equation  the  complete m o l e c u l a r  as  This  from s e d i m e n t a t i o n that this  (44), thereby  weight d i s t r i b u t i o n  the  Numerals i n parentheses  refer  weight  obtained  limitation  allowing  (MWD)  to  be  Thus,  applied to  to the  was  solution  e q u i l i b r i u m data.  a p p r o a c h c o u l d be  the  Until recently,  w e i g h t s c o u l d be  technique  being  of a l l o w i n g  when a p r a c t i c a l m a t h e m a t i c a l  was  felt  advantages of  t o come t o e q u i l i b r i u m .  of t h i s  is  of p r o t e i n - p r o t e i n  amount o f i n f o r m a t i o n , s u c h  o v e r c o m e i n 1968  was  the  thermodynamic t h e o r y ,  z average molecular  determined  study  T h i s a p p r o a c h has  system under study  through  equilibrium ultracentrifugation  p o t e n t i a l method f o r t h e  interactions.  *  ultracentrifugation  E a c h o f t h e s e methods  of fluorescence p o l a r i z a t i o n ,  Sedimentation  it  on  conventional  Johnston-Ogston e f f e c t  evaluation of accurate case  of  velocity  fluorescence polarization.  studied.  (9,10,27,34,35,36)*  m a c r o m o l e c u l a r i n t e r a c t i o n s made u s e methods s u c h  necessitates  the  literature  cited.  study of food p r o t e i n s and t h e i r  interactions.  T h i s t h e s i s focused on two major o b j e c t i v e s . initial (UV)  aim was  t o put i n t o o p e r a t i o n an  The  ultra-violet  o p t i c a l system newly a c q u i r e d f o r our L2-65B  p r e p a r a t i v e u l t r a c e n t r i f u g e and to i n t e g r a t e t h i s w i t h a data a c q u i s i t i o n system and a desktop  unit  computer.  These components were o b t a i n e d t o allow f o r the automation of m o l e c u l a r weight d e t e r m i n a t i o n s . t i v e was  t o apply the MWD  The  second o b j e c -  c a l c u l a t i o n to p r o t e i n s ,  u s i n g model systems and p r o t e i n mixtures  to evaluate  i t s p o t e n t i a l u s e f u l n e s s i n a s s e s s i n g component concentrations.  LITERATURE REVIEW Many macromolecules  such as s y n t h e t i c polymers and  biopolymers, are produced i n v a r i o u s chain lengths and are, t h e r e f o r e , heterogeneous i n m o l e c u l a r weight. The measurement o f t h i s h e t e r o g e n e i t y i s o f fundamental importance i n macromolecular chemistry because the m o l e c u l a r weight d i s t r i b u t i o n o f a polymer i s r e l a t e d t o i t s end use c h a r a c t e r i s t i c s o r performance.  A practical  example o f such a r e l a t i o n s h i p has been p r o v i d e d by G e h a t i a and W i f f  (21) who d i s c u s s e d how polymers made  up o f d i f f e r e n t MWDs c o u l d a f f e c t the performance o f d e c e l e r a t o r parachutes f o r space v e h i c l e s and h i g h speed aircraft.  I n the case o f b i o p o l y m e r s , t h e MWD o f  dextran was o f i n t e r e s t t o W i l l i a m s and Saunders  (65,66) ,  s i n c e t h e r e appeared t o be a c l o s e r e l a t i o n s h i p between the m o l e c u l a r s i z e o f the dextran molecules and t h e i r r e t e n t i o n time i n the body, when dextran was used as a plasma extender.  A n a l y s i s o f the MWDs o f polymers i s  not r e s t r i c t e d t o d e t e r m i n i n g end use c h a r a c t e r i s t i c s . Determination o f MWDs c o u l d a l s o be u s e f u l i n s t u d i e s of macromolecular a s s o c i a t i o n s and i n t e r a c t i o n s P r o t e i n s , which are u s u a l l y homogeneous, q u i t e  (2 4). often  s e l f - a s s o c i a t e and thereby become p o l y d i s p e r s e i n terms of m o l e c u l a r weight.  5.  T h e r e f o r e , p r o t e i n mixtures can  be  studied  m o l e c u l a r i n t e r a c t i o n s to e l u c i d a t e and  f o r macro-  the r e a c t i o n mechanisms  t o c a l c u l a t e the a s s o c i a t i o n c o n s t a n t s  (43).  In order to e v a l u a t e the p o l y d i s p e r s i t y o f a macrom o l e c u l a r system, most methods e i t h e r separate or d i s t r i b u t e the  sample.  In g e n e r a l ,  complete  re-  separation  i s not achieved so t h a t computational methods are t o e l u c i d a t e the The  required  s i z e d i s t r i b u t i o n i n the o r i g i n a l sample.  u l t r a c e n t r i f u g e was  one  o f the  first  instruments used  to e v a l u a t e the p o l y d i s p e r s i t y of macromolecular systems. S p e c i f i c a l l y , sedimentation-diffusion c e n t r i f u g a t i o n has  (19).  are p l a c e d  the p r i n c i p l e s o f thermo-  In t h i s method of a n a l y s i s , the macromolecules  i n a moderate c e n t r i f u g a l f i e l d  concentration  gradient  M  calculated using  the  2RT app  (1 -  where:  forming a  t h a t i s measured o p t i c a l l y .  these data the m o l e c u l a r weight o f an can be  ultra-  been of p a r t i c u l a r i n t e r e s t because of  i t s t h e o r e t i c a l f o u n d a t i o n based on dynamics  equilibrium  From  i d e a l macromolecule  following r e l a t i o n  (11):  din c Eq.  vp) 2  1  w  apparent m o l e c u l a r weight o f the  solute  (daltons) R  u n i v e r s a l gas deg.  - mole)  constant  (8.314 x 10  ergs/  6.  T  - temperature  P  - d e n s i t y o f the s o l v e n t  v  - p a r t i a l s p e c i f i c volume o f the s o l u t e  0)  - a n g u l a r v e l o c i t y (rad/s)  c  - c o n c e n t r a t i o n o f the s o l u t e (g/1)  r  - r a d i a l d i s t a n c e from c e n t e r o f r o t a t i o n  (°K) (g/ml)  (cm)  For an homogeneous i d e a l s o l u t e , a p l o t o f the n a t u r a l l o g a r i t h m o f c o n c e n t r a t i o n v e r s u s r a d i a l d i s t a n c e squared w i l l g i v e a s t r a i g h t l i n e w i t h the slope d i r e c t l y p r o p o r t i o n a l t o the m o l e c u l a r weight o f the s o l u t e .  In the  case o f i d e a l p o l y d i s p e r s e systems of c h e m i c a l l y molecules, a c u r v i l i n e a r p l o t i s o b t a i n e d .  identical  Various  m o l e c u l a r weight averages such as the number average weight average  (M ) w  and z average  (M ), n  (M^) can be c a l c u l a t e d  by s e v e r a l procedures ( 1 9 , 2 9 , 3 9 , 6 4 ) .  These m o l e c u l a r  weight averages g i v e l i m i t e d i n s i g h t i n t o the f r a c t i o n a l d i s t r i b u t i o n o f the component molecules o f the system. In p r i n c i p l e , s e d i m e n t a t i o n e q u i l i b r i u m  ultracentrifugation  i s capable o f p r o v i d i n g the e n t i r e m o l e c u l a r weight d i s t r i b u t i o n , as w e l l as the weight average m o l e c u l a r weights.  7.  The p o s s i b i l i t y o f o b t a i n i n g the MWD  from the  c o n c e n t r a t i o n g r a d i e n t formed by e q u i l i b r i u m u l t r a c e n t r i f u g a t i o n was p o i n t e d out by Rinde i n 1928  (44).  He developed the f o l l o w i n g r e l a t i o n s h i p on which most of the methods f o r o b t a i n i n g MWDs from u l t r a c e n t r i f u g a l data are based.  f  c(£) c  XM exp(-XM?) 1 - exp(-XM)  o  where:  f(M)dM  Eq. 2  X  - i s a complex f u n c t i o n o f r o t o r speed  £  - a dimensionless  radial  f(M) - d i f f e r e n t i a l molecular c  o  - initial  coordinate weight d i s t r i b u t i o n  concentration  c(£) - c o n c e n t r a t i o n a t some r a d i a l M  - molecular  weight  distance  (daltons)  P r i o r t o the completion of Rinde*s d i s s e r t a t i o n , Svedberg and N i c h o l s  (53) attempted t o o b t a i n a MWD  Rinde's i n t e g r a l r e l a t i o n . on simultaneous equations, t a i n i n g negative realistic Lansing  However, t h e i r approach, based r e s u l t e d i n a s o l u t i o n con-  c o n c e n t r a t i o n v a l u e s t h a t were not  i n terms of the p h y s i c a l s i t u a t i o n .  and Kraemer  molecular  using  (2 9) demonstrated how  weight averages M^ and M .  In  1935,  t o o b t a i n the  They showed t h a t  8.  the t h r e e m o l e c u l a r weight averages, M ,  M  n  be used t o c a l c u l a t e the probable MWD, MWD  and M , z  assuming  f o l l o w e d a l o g normal d i s t r i b u t i o n .  which attempted  w  This  could  t h a t the  approach,  to f i t m o l e c u l a r weight moments t o an  assumed d i s t r i b u t i o n , and o t h e r s i m i l a r approaches were l a t e r shown t o be i m p r a c t i c a l associates  (28,67).  Wales and h i s  (55-58) t r i e d t o a v o i d the f i x e d  forms o f L a n s i n g and Kraemer  functional  (29), and used osmotic  p r e s s u r e d a t a to generate the osmotic v i r i a l  coefficient  f o r c o r r e c t i o n of nonideal e f f e c t s . More r e c e n t l y , t h e r e has been a resurgence o f i n t e r e s t i n o b t a i n i n g a p r a c t i c a l s o l u t i o n t o Rinde's e q u a t i o n . Sundelof  (52) proposed a method, l a t e r r e f i n e d by  Provencher  (41), based on the F o u r i e r c o n v o l u t i o n theorem.  Provencher and Gobush (42) a l s o suggested a combination o f quadrature and l e a s t squares to s o l v e the Rinde e q u a t i o n , an demonstrated  h i s approach by g e n e r a t i n g s y n t h e t i c d a t a  from a Wesslau d i s t r i b u t i o n , and then u s i n g a S c h u l t z f u n c t i o n t o make h i s f i t .  A l l o f the methods t h a t  attempted t o s o l v e the Rinde e q u a t i o n f o r the frequency v a l u e s f ( M ) , s u f f e r e d from the problem encountered by Svedberg  originally  and N i c h o l s (53); they produced  weight f r a c t i o n s t h a t were n e g a t i v e .  T h i s problem  overcome independently w i t h t o t a l l y d i f f e r e n t by Donnelly  (14,15) and S c h o l t e (47,48).  was  approaches  9.  Donnelly  (14) r e c o g n i z e d t h a t the c o n c e n t r a t i o n  d i s t r i b u t i o n o f polymeric s o l u t e s was L a p l a c e transform.  He was  i n the form of a  a b l e t o c o n v e r t the Rinde  equation t o a Laplace t r a n s f o r m and o b t a i n the MWD its  inverse.  The method was  from  theoretically rigorous,  worked w e l l f o r unimodal d i s t r i b u t i o n s and had  the  advantage of r e q u i r i n g data from o n l y one r o t o r speed. However, i t was and was  r e s t r i c t e d to continuous  unable to handle multimodal  Scholte  (47) proposed  equation-was transformed  systems  (15).  a method whereby the Rinde  to a s e t of l i n e a r  By assuming a s e r i e s of m o l e c u l a r weights e x p o n e n t i a l l y spaced,  distributions  equations.  t h a t were  f(M) became the o n l y unknown t o be  s o l v e d f o r i n the Rinde equation.  Through the use o f  l i n e a r programming with p o s i t i v e c o n s t r a i n t s , a f e a s i b l e s o l u t i o n c o u l d be obtained.  S c h o l t e ' s method r e q u i r e d  the use of s e v e r a l r o t o r speeds to p r o v i d e p a r t i a l fractionation  of the sample, and thus the d i s t r i b u t i o n  f o r multimodal  systems.  More r e c e n t l y , Gehatia and W i f f developed  (22,23,60-63)  a s o p h i s t i c a t e d mathematical  data t o p r o v i d e MWDs.  They r e c o g n i z e d the f a c t t h a t  the Rinde equation i s an Improperly as d i d other workers c o u l d be circumvented  treatment of the  Posed Problem  (IPP),  (30), and r e a l i z e d t h a t t h i s problem by a p p l y i n g Tikhonov's  regularization  10.  function. IPP  The  r e c o g n i t i o n t h a t the Rinde equation  explained  the problems encountered i n o b t a i n i n g a  d i s t r i b u t i o n from the Rinde equation. (21)  Gehatia  and  found t h a t r e g u l a r i z a t i o n alone c o u l d not  more complex d i s t r i b u t i o n s .  They then i n c o r p o r a t e d (48)  used q u a d r a t i c programming  techniques  i n theory,  (63).  These  and  the  later  the a n a l y s i s o f unimodal to pentamodal  d i s t r i b u t i o n s , but i n p r a c t i c a l terms, t r i m o d a l a t one  Wiff  resolve  method of l i n e a r programming of S c h o l t e  allowed,  i s an  speed were probably  systems  the l i m i t t h a t c o u l d be  analyzed.  T h i s very s o p h i s t i c a t e d technique i s unusable f o r most researchers Wan  due  (59) e v a l u a t e d  u s i n g dextran Donnelly  to the e x c e s s i v e  was  the methods o f Donnelly  as a p o l y d i s p e r s e  nonideal  sample.  and  Scholte,  The method o f  found to be more convenient as the c a l c u l a t i o n s  were somewhat simpler and required.  computer time r e q u i r e d .  o n l y one  run a t one  speed  was  L a t e r , Adams e t a l (1,2,50) showed t h a t systems c o u l d be s t u d i e d by u s i n g the  s c a t t e r i n g second v i r i a l c o e f f i c i e n t Magar (30)  light  B^ . g  reviewed the v a r i o u s methods f o r o b t a i n i n g  MWDs from a s t a t i s t i c i a n ' s p o i n t of view. use of a l e a s t squares method  He  suggested  the  CL^ norm) to be a p p l i e d to  S c h o l t e ' s method, as a s u b s t i t u t e f o r l i n e a r programming. T h i s was  considered  t o be a b e t t e r method s i n c e there were  s t a t i s t i c a l parameters a v a i l a b l e to e v a l u a t e  the f i t of  the  11.  distribution.  A l i m i t a t i o n of the  norm, however, was  the p o s s i b l e appearance o f n e g a t i v e weight making the method u n r e a l i s t i c . the method of s t e e p e s t descent of Nelder and Mead  I f the  fractions, norm  failed,  (31) o r the simplex method  (37). w i t h p o s i t i v e c o n s t r a i n t s were  a l t e r n a t i v e s to'be attempted.  Magar a l s o  the use o f curve f i t t i n g techniques  recommended  ( I ^ norm o r o r t h o g o n a l  polynomials) f o r smoothing t h e raw u l t r a c e n t r i f u g a l d a t a . To date no r e s e a r c h e r s have r e p o r t e d the a p p l i c a t i o n o f the recommendations made by Magar.  12.  PART I .  MOLECULAR WEIGHT DETERMINATIONS  THEORY  Sedimentation equilibrium experimental technique that  u l t r a c e n t r i f u g a t i o n i s an  facilitates  m o l e c u l e b e h a v i o r under t h e i n f l u e n c e gravitational  force.  between t h e f o r c e s force, of  The b a s i c  study o f macro-  o f a moderate  concept involves  a balance  o f s e d i m e n t a t i o n , due t o c e n t r i f u g a l  and d i f f u s i o n , due t o t h e d e v e l o p m e n t o f a  lowered p o t e n t i a l .  By c h o o s i n g  a r e d i s t r i b u t i o n of the solute approach e q u i l i b r i u m kept constant. attained,  i f rotor  Although true  after a period  concentration  an a p p r o p r i a t e  occurs that  speed,  t e n d s t o :: •  speed and temperature a r e equilibrium  o f time,  will  n e v e r be  t h e changes i n t h e  d i s t r i b u t i o n become n e g l i g i b l e when  measured by t h e o p t i c a l system. distribution  region  of a solute  amount o f i n f o r m a t i o n ultracentrifugation  The  concentration  at equilibrium  about t h e system.  a l s o has a d i s t i n c t  o t h e r methods o f s t u d y i n g  can provide a  large  Equilibrium advantage  over  the behavior o f macromolecules  in  s o l u t i o n b e c a u s e t h e method h a s a t h e o r e t i c a l  on  the f i r s t  and s e c o n d laws o f t h e r m o d y n a m i c s .  foundation The  theory of u l t r a c e n t r i f u g a t i o n i s b a s i c a l l y the a p p l i c a t i o n of  t h e r m o d y n a m i c s t o s y s t e m s u n d e r an e x t e r n a l  In  the case o f e q u i l i b r i u m  force.  u l t r a c e n t r i f u g a t i o n , there a r e  13.  no  kinetic  avoiding  quantities  due  t o s o l u t e movement,  thus  the c o m p l i c a t i o n s of t r a n s p o r t theory.  thermodynamic approach a l l o w s equations  capable  f o r the  of d e s c r i b i n g very  The  formulation of complex systems t h a t  are p r e s s u r e dependent o r p o l y d i s p e r s e i n m o l e c u l a r or  in partial  specific  Equilibrium sociated with The  providing  volume.  u l t r a c e n t r i f u g a t i o n has  l o n g been  the determination of molecular  theory developed solutions  over  the y e a r s  has  led to  f o r the g e n e r a l case,  partial  specific  the theory e x i s t s work on  the d e t e r m i n a t i o n of m o l e c u l a r  including  weights Although  i n general weights  b e e n done w i t h r e l a t i v e l y  the  most  via ultra-  simple  systems,  w i t h ones w h i c h a l l o w a s s u m p t i o n s f o r s i m p l i f y i n g  complex t r e a t m e n t . the  'basic'  The  An  velocity  equations  f o r sedimentation  presented.  equilibrium  ultra-  total  g r a d i e n t a t any the c e l l  flow equation  the premise  radial  b o t t o m has  p o s i t i o n between t o be  zero.  f o r sedimentation  q s o l u t e s as g i v e n by  Fujita  of  the  system i s c o n s t a n t , o r t h a t the  potential  on  be  from  total  of the  the  equation  t h a t the  potential  with  equilibrium  flow equation w i l l  c e n t r i f u g a t i o n are based  m e n i s c u s and  the  i l l u s t r a t i o n of the d e r i v a t i o n  sedimentation  sedimentation  for  equations  of n o n i d e a l i t y .  f o r complex a n a l y s e s ,  c e n t r i f u g a t i o n has or  v o l u m e s and  as-  weights.  i n f l u e n c e of pressure, heterogeneous molecular and  weight  the  Starting  i n the c e n t r i f u g e  (18), the f o l l o w i n g  r e l a t i o n describes  the p o t e n t i a l g r a d i e n t  in a centrifugal  field:  Sy — i 6r where:  „ =  fi -  /Sy \ £ j=l\6c./  6c . — 6r  4  v, p ) u ^ r  -  Eq.  1  p  3  q  (k=0,l...,q, m=l,2...,q) - t o t a l number o f s o l u t e s  y^  - chemical p o t e n t i a l per gram o f component  P  - hydrostatic pressure  (torr)  - p a r t i a l s p e c i f i c volume of component k (unhydrated) E q u a t i o n 3 i s based on two  commonly made assumptions;  system i s independent of p r e s s u r e the  solute i s a nonelectrolyte.  (incompressible),  and  In p r a c t i c e , these  assumptions are g e n e r a l l y v a l i d ,  since  sedimentation  e q u i l i b r i u m i's commonly a t t a i n e d a t r e l a t i v e l y low and  the charge e f f e c t s can be n u l l i f i e d due  o f the charge by electrolyte.  'swamping'  a d d i t i o n of a s m a l l amount of  supporting  Equation 3 b a s i c a l l y describes  how  the  (y) of the system changes w i t h  radial  (r) as the components are sedimented under a  centrifugal force.  The  f i r s t term on the r i g h t hand s i d e  o f the e q u a t i o n r e p r e s e n t s on the k components. side contains  The  the  sedimentation forces  i s no net t r a n s p o r t  At and  acting  second term on the r i g h t hand  a summation r e p r e s e n t i n g  f o r c e of d i f f u s i o n .  the two  speeds  to  chemical p o t e n t i a l distance  the  the  counteracting  sedimentation e q u i l i b r i u m no p o t e n t i a l g r a d i e n t ,  terms can be equated as  follows:  there  therefore,  15.  - 2 V(1 - v p ) u r =2_jV±] i  -  k=l  k  d C  dr  k  d  Eq. 4 (i=l,2...,q)  where:  |  Eq. 5  5p \ i  ^ik  k'  6 c  T,P,c  Equation 4 represents  m  q f i r s t order d i f f e r e n t i a l  t h a t can be s o l v e d f o r q s o l u t e c o n c e n t r a t i o n s of r a d i a l d i s t a n c e  (r) .  as f u n c t i o n s  T h i s c e z i e r a l i z e d form o f the  equation i s r a t h e r d i f f i c u l t t o s o l v e . a much simpler  equations  two component system  By s t a r t i n g w i t h  ( s o l v e n t and s o l u t e ) ,  E q u a t i o n 4 reduces t o :  (1 - v  l P  )  2 W  r =  P  l  ^1  E  q.  6  dr U t i l i z i n g the assumption t h a t the s o l u t e i s a n o n e l e c t r o l y t e , the chemical p o t e n t i a l can be w r i t t e n :  , o, U  l  =  {  U  l  }  c  RT  ± +  ln(y c )  — M  where:  Eq. 7  l  (u°) - i s the r e f e r e n c e p o t e n t i a l p e r gram o f 1 c s o l u t e 1 so t h a t t h e a c t i v i t y c o e f f i c i e n t on the c - c o n c e n t r a t i o n  s c a l e approaches  u n i t y as c^ tends t o zero Y^  - a c t i v i t y c o e f f i c i e n t of solute 1  16..  Substituting  M^(l  Equation  V-^p)w  -  7 into  6  yields:  1 +  Eq.  c  RT  The  dc.  61ny-  rc  6c,  J  T,P  8  dr  r i g h t hand s i d e o f t h e e q u a t i o n r e p r e s e n t s t h e n o n i d e a l  c o n t r i b u t i o n s due of the  solute  to concentration.  i s low  i d e a l behavior,  enough and  this  term  ignored.  The  natural  efficient  i s a complex  concentration Taylor  and  the  reduces  I f the concentration solute  to unity  thereby  expresses  and  be  can  l o g f u n c t i o n of the a c t i v i t y  co-  f u n c t i o n of molecular weight  c a n be  expressed  a s an e x p a n s i o n  and of  a  series:  2  lny  = i i i M  Substituting  B  c  this  +  M  2  l 2 l B  c  Eq.  +  expansion  term  back i n t o E q u a t i o n  9  8  produces: M  n  (1 -  v p ) to r c 1  1 +  (BjM^  + V )c 1  1  +  RT  (2B  dc 2 M l  B  i iV ^i M  ) c 2  dr  T h i s equation provides the b a s i s f o r the analysis  Eq.  +  thermodynamic  of sedimentation e q u i l i b r i u m data  solutions.  for binary  I f t h e c o n c e n t r a t i o n o f t h e component  under  10  17.  study  i s s u f f i c i e n t l y h i g h t o cause the n o n i d e a l i t y term  to become s i g n i f i c a n t , e v a l u a t i o n o f the n o n i d e a l i t y coefficient, to express  B, i s r e q u i r e d .  i d e a l behavior  c o n c e n t r a t i o n s are used Equation  In g e n e r a l , p r o t e i n s tend  i f c o r r e c t c o n d i t i o n s and low  (46). Under these c o n d i t i o n s  10 can' be f u r t h e r s i m p l i f i e d t o :  __i  V  =  1  -y  dr  ^  ^  y  Eq  . ii  ..RT  T h i s equation can then be transformed  i n t o the f o l l o w i n g  e x p e r i m e n t a l l y u s e f u l form:  d  l  n  C  d(r ) 2  l  =  M  l (  l - v )co  2  l P  E  a p p l i c a b l e t o the d e t e r m i n a t i o n  2  weights.  r a d i a l distance  i s d i r e c t l y p r o p o r t i o n a l t o molecular  The m o l e c u l a r weight o b t a i n e d  weight.  i s o n l y an apparent v a l u e  s i n c e the e f f e c t o f c o n c e n t r a t i o n has n o t been taken account.  1  i s directly  of molecular  s l o p e o f a p l o t o f In absorbance v s .  squared  >  2RT  When UV o p t i c s a r e used, the above equation  The  q  A range o f c o n c e n t r a t i o n s c o u l d be run t o  determine the presence o f c o n c e n t r a t i o n dependence.  into  18.  EXPERIMENTAL  Ultracentrifugation  The is  standard c e n t r i f u g e used  for analytical  t h e Beckman M o d e l E u l t r a c e n t r i f u g e  sophisticated laboratory; centrifuge  and  on  not a v a i l a b l e  i n our  equipped used.  principles  w i t h a n e w l y m a r k e t e d UV The  d e s i g n o f t h e UV  employed  i n t h e M o d e l E UV  p r o v i d e d t h e L2-65B u n i t w i t h a n a l y t i c a l on  v a r i a b l e wavelength,  sector scanning  absorbance  range  L2-65B v e r s i o n , c o m p a r a b l e and expensive The  UV  but  - 2.0  the  scanner  as c e l l s  t h e Model E.  capabilities. such and  as an  alternative  this  scanner  was  t o i t s more  accessory c o n s i s t e d o f t h r e e major  inside and  chamber o p t i c a l  unit,  the e l e c t r o n i c s u n i t  and  c o u n t e r b a l a n c e s were t h e  The  s c a n n i n g a c c e s s o r y was  thereby measuring  of the s o l u t e  t h e amount o f UV  the  door  (multiplexer).  equipment n e c e s s a r y t o run the  concentration distribution cell,  Scanner,  were n o t p r e s e n t i n  i n other r e s p e c t s the  an e c o n o m i c a l  scanning  remaining  such  o f 0.0  the Model E,  was  counterpart.  components, mounted  single  scanning  accessory  Numerous f e a t u r e s a v a i l a b l e  The  This  however, an L2-65B p r e p a r a t i v e u l t r a -  a c c e s s o r y was based  i n s t r u m e n t was  (4).  work  centrifuge,  same a s f o r  able to scan i n the  the  centrifuge  radiation  19.  absorbed was  a  at  various  short  arc  wavelengths, ference rotor  around The  through  formed  at  the  the  by  a  the  image.  and  reference  signals  sectors  strip  chart  worked with  tube  the  rotor  The  multiplexing  of  rotor.  the  gave  a  cell  at  trace  The of  in  one  rather  as  unit  edges  the an  the  a  from  advantage was  optical  scan sample  from  where  the  recorded  on  therefore,  scanning  ring  on  of  the  called  1.0  a  absorbance  2).  in  The the  the  image.  UV  This  'scan', the  signal  scanning  other form  multi-  bottom  distribution in  over  a  chopper.  recording,  a  the  spectrophotometer,  beam  coded  (Figure  being tube.  to  the  passed  allowed  with  and  carriage  able  unit,  beamed  chart  output  cell  difference  also  along  of  a multiplexer  light  spinning  1)  was  were  concentration  major  than  that  of  inter-  the  a mobile  scanning  a  an  band  Figure  received  double  strip  the  that  a  narrow  source  photomultiplier  on  cell  deciphering  reference  had  systems, signal  by  equilibrium,  cell  system  acting  the  the  The  e s s e n t i a l l y as  rotors  the  light  onto  image  motor  and  recorder.  placed  to  through  compared  the  signals of  a  system,  mounted  speed  which  directed  with  was  The  i s o l a t e d by  was  slit  Electrical  were  was  optical  cell,  entrance  photomultiplier  and  of  constant  from  nm,  light  photomultiplier  driven  positions.  lamp  280  schematic  passed  The  mercury  filter. (see  radial  of  optical an  signal  electrical was  20.  light  source  p h c t a r o i l t i p l i e r tube  Slit  0  BBSS I ,spherical mirror  light path  \  I  \ I  Figure 1.  rotor  Schematic diagram of the UV o p t i c a l system.  H  Figure 2.  Recorder trace (scan) of a typical seclimentation equilibrium experiment using a double sector c e l l .  A - 1.0 absorbance signal  F - a i r - solvent meniscus  B - outer reference edge  G - a i r baseline  C - counterbalance darkspace  H - counterbalance darkspace  D - solute distribution pattern  I - inner reference edge  E - solution - solvent meniscus  amenable t o d i g i t i z a t i o n and t h e r e f o r e t o automated processing  data  techniques.  P r i o r t o any u l t r a c e n t r i f u g a l work, the o p t i c a l system o f the u l t r a c e n t r i f u g e was compared t o a s p e c t r o photometer f o r l i n e a r i t y .  T h i s was done u s i n g a c a r e f u l l y  d i l u t e d s e r i e s o f c o l c h i c i n e s o l u t i o n s t o produce a standard  curve o f c o n c e n t r a t i o n  v s . absorbance.  i n d i v i d u a l d i l u t i o n s were then loaded  These  i n t o the c e n t r i f u g e ,  spun a t 12,000 rpm and the absorbance recorded.  The  p r e p a r a t i v e UV scanner response was found t o be l i n e a r up t o an absorbance o f 1.2. found t o d i f f e r by a constant  The absorbance v a l u e s were t h a t was e s s e n t i a l l y  e q u i v a l e n t t o the c o r r e s p o n d i n g l y the c e n t r i f u g e c e l l .  A check o f the i n t e r f e r e n c e  showed t h a t i t . t r a n s m i t t e d of wavelengths, with  l o n g e r path l e n g t h i n filter  only a r e l a t i v e l y narrow band  a maximum i n t e n s i t y a t 2 78 nm.  Other  checks, such as r o t o r wobble a t low speeds, o i l d e p o s i t i o n from the d r i v e , b a s e l i n e c o n s i s t e n c y ,  and r e p r o d u c i b i l i t y  of the r e f e r e n c e edges, c e l l bottom and meniscus t r a c e s , were a l s o made, and shown to be w i t h i n t o l e r a n c e .  Special  a t t e n t i o n was p a i d t o the c l e a n l i n e s s o f the o p t i c a l system with run.  a l l components o u t s i d e the housing c l e a n e d  a f t e r every  The o p t i c a l tower was s e a l e d t o p r e v e n t a d i f f u s i o n  of o i l onto the m i r r o r s , so t h a t the u n i t r e q u i r e d disassem bly  o n l y a f t e r extended p e r i o d s  motor was purchased t o  o f time.  An a d d i t i o n a l  drive  the scanner  data a c q u i s i t i o n The  system  speed,  A complete  a l l o w i n g the  t o r e c o r d more d a t a p e r s c a n .  available  list  proteins of a n a l y t i c a l  c a n be f o u n d  i n the Appendix,  o f t h e s e p r o t e i n s were c h e c k e d  acrylamide d i s c  the buffer  a p p r o p r i a t e absorbance u s i n g a Beckman DB  (13).  for at least  t h e y were d i s s o l v e d values  grade. S e c t i o n A. by p o l y -  In a l l cases,  24 h o u r s  i n , then d i l u t e d (280 nm)  against to  t h a t were m e a s u r e d  spectrophotometer.  For molecular weight  distribution  and l o a d i n g p r o c e d u r e s  sector c e l l s .  studies,  the  cell  d e s c r i b e d by Chervenka  were f o l l o w e d , u s i n g r e g u l a r 12 mm r e s i n double  for purity  gel electrophoresis  t h e p r o t e i n s were d i a l y s e d  assembly  thus  p r o t e i n s s t u d i e d i n t h i s work were a l l  commercially  All  a t a slower  carbon-filled  (8)  epoxy  Fluorocarbon solution  was  added t o t h e s e samples t o a l l o w a c c u r a t e d e t e r m i n a t i o n of t h e c e l l utilized for  bottom.  The t h r e e p l a c e Y p h a n t i s  f o r simple molecular weight  a leakage  problem  described  i n t h e A p p e n d i x , S e c t i o n B, was  Normally,  cell,  a black anodized  was  used  which h e l d three c e l l s  All  r u n s were p e r f o r m e d  being monitored  procedure,  followed.  four place rotor  (AN-F:>  counterbalance.  a t 20°C w i t h t h e  by an i n f r a r e d  In order t o avoid  a special  and a  was  d e t e r m i n a t i o n s and  a s s e s s i n g c o n c e n t r a t i o n dependency. with this  cell  sensitive  temperature radiometer  located to  under  the rotor.  The speed  b e r u n was e s t i m a t e d f r o m  molecular  weight  on page  f o rthe Analytical  running  speed  was  from  several  scans  were  maximum  material.  t o running  for  use as the true  was  required  tortion  Data  equilibrium  had elapsed, and compared  and a f i n a l  the rotor  scan  was  accelerated  the rotor  was  s c a n was  taken  The d e c e l e r a t i o n results  at high  was  of a l l protein  and a f i n a l  erroneous  o f t h e q u a r t z windows  caused  procedure by  dis-  speeds.  Processing  One  o f the objectives  recently  eliminate data  speed  five  had occurred, t h e system  was c o m p l e t e d ,  baseline.  t o avoid  time  intervals,  to deplete the cell  When d e p l e t i o n  decelerated  a  taker, a t h o u r l y  a r u n was c o m p l e t e d ,  speed  t o reach  the calculated  c o n s i d e r e d t o be a t e q u i l i b r i u m  to  actual  o f Van Holde and  I f no o b s e r v a b l e changes  When  ( 8 ) . The  The time  (54) , a n d a f t e r  taken.  Manual o f  t h e odometer.  the calculation  was  o f rpm v s .  by a v e r a g i n g a t l e a s t  Baldwin  was  Centrifuge  a sample  was d e t e r m i n e d  estimated using  visually.  the graph  45 o f C h e r v e n k a ' s  Methods  readings  a t which  acquired t h e manual  points  from  of this  s t u d y was t o  data acquisition procedures  t h e UV  scanner  system,  involved strip  utilize  i n order t o  i n obtaining  chart  output.  D i g i t i z e d data obtained and s t o r e d by the data  acquisition  system c o u l d then be processed through the use o f a desktop computing system.  T h i s system r e p l a c e d the r a t h e r  t e d i o u s c a l c u l a t i o n s a s s o c i a t e d w i t h molecular weight d e t e r m i n a t i o n s with an automated, r o u t i n e procedure, i n c r e a s i n g the accuracy and r e d u c i n g human e r r o r .  A.  Hardware  1.  Data a c q u i s i t i o n system  The data a c q u i s i t i o n system used i n t h i s  study  c o n s i s t e d o f three components, a Schlumberger S o l a t r o n data t r a n s f e r u n i t , a and a Texas Instruments terminal  S o l a t r o n A220 d i g i t a l  voltmeter  S i l e n t 700 ASR e l e c t r o n i c  data  ( t e l e t y p e ) w i t h a c a s s e t t e tape a c c e s s o r y . A l l  t h r e e components were mounted on a l a b o r a t o r y c a r t t o make t h e u n i t s e l f - c o n t a i n e d and mobile  (Figures 3 & 4 ) .  The data t r a n s f e r u n i t had a time  controllable  m u l t i c h a n n e l c a p a b i l i t y and served t o r e c e i v e and d i g i t i z e the incoming v o l t a g e s i g n a l .  The d i g i t a l  voltmeter  d i s p l a y e d t h e v o l t a g e and t r a n s f e r r e d the data t o the teletype.  The t e l e t y p e i n t u r n p r i n t e d a hard copy o f  the d i g i t i z e d v a l u e s on heat s e n s i t i v e paper, ,and s i m u l t a n e o u s l y recorded the data on magnetic tape.  The  F i g u r e 3.  Photograph o f t h e d a t a a c q u i s i t i o n system on a l a b o r a t o r y c a r t - t e l e t y p e , v o l t m e t e r and d i g i t i z e r .  Figure 4.  Photograph of the acquisition system, multiplexing unit and ultracentrif uge.  28.  t e l e t y p e was  a b l e t o i n t e r f a c e d i r e c t l y to a Monroe  programmable desktop c a l c u l a t o r t h a t was  1880  used t o p r o c e s s  the data, i f p r o v i d e d w i t h the a p p r o p r i a t e program. schematic  A  diagram of the data p r o c e s s i n g o p e r a t i o n i s  presented i n F i g u r e 5.  2.  Desktop computation  and  plotting  Most of the c a l c u l a t i o n s c a r r i e d out i n t h i s work were programmed on a Monroe 1880 Calculator.  Scientific  The b a s i c u n i t came w i t h 512  and 64 main data r e g i s t e r s .  Printing  program steps  F o r t h i s work, the program  step c a p a c i t y and memory were expanded t o the maximum of 4,096 and  512 r e s p e c t i v e l y .  The c a l c u l a t o r had  a magnetic  c a r d reader f o r preprogrammed i n s t r u c t i o n s or s t o r e d d a t a . The p r i n t i n g u n i t allowed a hard copy o f the  calculation  r e s u l t s to be o b t a i n e d and c o u l d a l s o l i s t programs t h a t had been w r i t t e n .  Most of the s c i e n t i f i c  f u n c t i o n s were  a v a i l a b l e and a low l e v e l programming language allowed programs t o be w r i t t e n on the b a s i s o f simple a l g e b r a i c logic.  Symbolic  a d d r e s s i n g , i n d i r e c t a d d r e s s i n g , sub-  r o u t i n e s , f l a g g i n g and d e c i s i o n c a p a b i l i t i e s were available.  With these f e a t u r e s , the Monroe 1880  cal-  c u l a t o r p r o v i d e d a v e r y powerful computing f a c i l i t y a t a low c o s t .  Centrifuge  S t r i p Chart Recorder  Digitizer and Voltmeter  Teletype Hard Copy  Magnetic Tape  Monroe 1880 Calculator Printer  F i g u r e 5.  Schematic diagram o f the data processing operations.  Plotter  30.  A p l o t t i n g u n i t was w i t h the Monroe 1880. be  controlled  or the  by  the  a l s o obtained to  T h i s Monroe PL-2  plotter  could  c a l c u l a t o r through a p l o t t i n g program  i n s e r t i o n of a p l o t t i n g r o u t i n e  that required  x-y  interface  the p l o t t i n g of d a t a .  i n t o the  The  programs  plotting  c a p a b i l i t y enhanced the u s e f u l n e s s of the  calculator  immensely i n u l t r a c e n t r i f u g a l work because a v i s u a l assessment of the data was further processing  3.  As was UV  scanner was  since  the  required  to d e c i d e  of the data a c q u i s i t i o n  system  centrifuge  mentioned p r e v i o u s l y ,  the  preparative  amenable to the data a c q u i s i t i o n  scanner produced a DC  v o l t a g e t h a t was  related  to the  absorbance d i f f e r e n c e  between the  sectors  of the  centrifuge  problem  cell.  w i t h scanner data o b t a i n e d by t h a t no  i n f o r m a t i o n was  c o o r d i n a t e f o r the problem was  One  directly two  associated  p r o v i d e d on  c a l i b r a t i o n of the  i n t e r v a l of the data t r a n s f e r i n the  unit.  scan was  was  r a d i a l d i s t a n c e , the  c a l c u l a t i o n of m o l e c u l a r weights.  overcome by  i n i t i a l point  system,  the a c q u i s i t i o n system  scanning motor movement i n r e l a t i o n to the  the  on  steps.  Application to the  often  x  This  c o n s t a n t speed s i g n a l pickup  In t h i s procedure,  i d e n t i f i e d and  a constant  31.  was  a p p l i e d to convert the time-based  data i n t e r v a l s t o  r a d i a l d i s t a n c e i n the c e n t r i f u g e c e l l .  In order t o  d e r i v e s u f f i c i e n t data p o i n t s per scan, a slower motor was  installed,  s i n c e the o r i g i n a l motor was  for velocity ultracentrifugation  B.  scanning designed •  experiments.  Software  A major programming e f f o r t was  undertaken  to allow  f o r complete p r o c e s s i n g of the raw data a v a i l a b l e from centrifuge.  Of the many programs w r i t t e n f o r the  the  cal-  c u l a t i o n s associated with u l t r a c e n t r i f u g a t i o n , only three w i l l be o u t l i n e d here, with the others b e i n g d e s c r i b e d i n the Appendix, S e c t i o n C. programs are of primary  briefly  These t h r e e  i n t e r e s t s i n c e they  illustrate  the a b i l i t y of the desktop c a l c u l a t o r t o automate the c a l c u l a t i o n of molecular  1.  weights.  Data c o n v e r s i o n program  T h i s program was  d i r e c t l y involved i n calculating  the n a t u r a l l o g of absorbance  (In A) and r a d i a l d i s t a n c e  2 squared  (x ) terms d i r e c t l y from the data p r o v i d e d by  the  a c q u i s i t i o n system, u t i l i z i n g the c o n s t a n t d e r i v e d from 2 the scanning motor.  The  In A vs. x  p a i r s were s t o r e d i n  r e s e r v e d r e g i s t e r s to a l l o w other programs t o work on  the  transformed  2.  Automatic  This from  raw  data.  linear  program  r e g r e s s i o n and p l o t t i n g  had the c a p a b i l i t y  a d e f i n e d memory r e g i s t e r  -routine This  and a l e a s t - s q u a r e s l i n e a r  program  weight  interval  was  useful  o f homogeneous  systems  from  of using  i n a  the  data  plotting  regression  f o r determining  program  procedure.  molecular  the constant  slope  2 of  the l n A vs.  visual the a  x  examination  data.  I f only  relationship.  of the stored  3.  Automatic  a portion of the data  and  be  into  using  only  that  with  back  feature  was  able  to retrieve  any d e s i r e d p o l y n o m i a l .  the polynomial  entered  repeated  linear,  m u l t i p l e r e g r e s s i o n program  program  f i t them by  provided  readily  was  f i tto  data.  calculation  This  allowed  of the least-squares linear  r e g r e s s i o n c o u l d be  part  The p l o t  coefficients  stored  data  The r e g r e s s i o n  which,  i n turn,  a back  calculation  subroutine  values  to generate  the best  that  could  used  2 the  original  their  x  derivatives.  evaluation  This  of molecular  program weights  was  useful  f i t data  or  f o r the  of heterogeneous  systems.  These t h r e e programs i n the Monroe 18 80 c a l c u l a t o r c o u l d be used t o convert the raw data coming from the a c q u i s i t i o n system i n t o m o l e c u l a r weights a minimum amount o f m a n i p u l a t i o n s .  with  Figure 6 i l l u s t r a t e s  the data flow i n and out of the programs d i s c u s s e d above.  RESULTS  As th? UV scanner was a newly marketed a c c e s s o r y , no p u b l i c a t i o n s had appeared  concerning i t s c a p a b i l i t i e s .  T h e r e f o r e , p r i o r t o attempting the more complex i n v e s t i g a t i o n o f p r o t e i n mixtures, the p r e p a r a t i v e UV scanning a c c e s s o r y and the u l t r a c e n t r i f u g e had t o be t e s t e d f o r accuracy i n e v a l u a t i n g m o l e c u l a r  Homogeneous  Systems  Experiments weights  weights.  were performed  t o determine  the molecular  of v a r i o u s standard p r o t e i n s and t o check f o r t h e  presence o f p r o t e i n c o n c e n t r a t i o n dependency. i n i t i a l experiments  These  were a l s o used t o t e s t and develop  the data a c q u i s i t i o n system, w i t h the c a l c u l a t i o n s both manually  performed  and by the automated system, i n o r d e r t o  e v a l u a t e the r e l a t i v e accuracy o f the two approaches. these experiments,  In  a l l o f the p r o t e i n s were r u n a t a wide  range o f c o n c e n t r a t i o n s ( u s u a l l y a s e r i a l d i l u t i o n ) t o  Linear Regression and P l o t t i n g . .  Raw  Data  Multiple Regression Program  Data Conversion Program  Slope (derivative)  Stored „ In A vs. x  Coefficients  Molecular Weight  Back Calculation  Derivatives  Figure 6.  Smooth  Molecular  In A vs. x"  Weights  Schematic diagram of data flow through the data conversion, linear regression and multiple regression programs.  determine  whether  dependent.  After  electrophoresis, conalbumin, for  of  ovalbumin  t h e UV  Nine  homogeneous  was  and t r y p s i n  will  serve system  by  polyacrylamide  inhibitor  linked  ovalbumin,  were  t o the data  system.  buffer,  concentrations of ovalbumin  ranging  i n absorbance,  were  i n c o n c e n t r a t i o n from loaded  into  three  a t approximately  The  time  was  t o reach  programmed  An  of  were  Section C). scanned  i n Figure  molecular  from  Once  values these  copy  chart,  was  produced  record  data  had been  and  entered  into  was  linear,  which  a linear  equation  the data  f o r each  (54)  being  system.  scan  can be  f o rthe calculation  position  and r a d i a l  accumulated,  values and  distance  they  were  r e g r e s s i o n program.  i t should  cells,  using the  the absorbance  the grid  t o l n absorbance  to  12,000 r p m .  acquisition  procedure  t o read  0.20  reaching equilibrium,  by t h e data  The normal  weights  the strip  these  7.  Upon  i n succession, with  and p u t on tape  example o f t h e hard  seen  calculated  v e r s i o n o f t h e Van Holde-Baldwin  (see Appendix,  digitized  equilibrium  0.05M, p H  i n  Yphantis  run i n a multiplace rotor  cells  obtained  the capabilities  and  all  chosen  The r e s u l t s  to illustrate when  concentration  proteins,  ultracentrifugation.  different  phosphate 0.60  four  scanning  processing  weight  assessing the purity  lysozyme  equilibrium  from  the molecular  convert squared.  plotted I f the plot  b e i f t h e s o l u t e was  pseudo-  DATE  - - -  12/12/76  RUN# SAMPLE  36 - - - - - - - - -  SCAN# O.D.  OVALBUMIN 3  _ _ _ _ _ _ _ _ _  0-5  -050  -050  -050  +449  -048  -050  -Q49  -049  -050  -050  -050  +695  +806  +806  +799  +806  +731  +597  +586  +185  +177  +167  +145  +128  +118  +101  +095  +081  +073  +064  +056  +048  +044  +036  +030  +023  +018  +317  -144  END  Figure  7.  OF  SCAN  Facsimile output.  #3  of the teletype  37 .  ideal  (20),  directly The  slope  of  the  proportional  to  molecular  above  samples the  the  procedures,  became v e r y  tedious  and  2  (din A/d(x weight  i f c a r r i e d out  a c q u i s i t i o n system,  out  plot  ))  was  then  (Equation  12).  manually  subject  to  for  several  error. be  Using  a l l procedures  could  carried  equilibrium  automatically. After  the  scans  of  the  samples  completed,  the  teletype  was  interfaced with  calculator  and  the  were  registers  of  the  data  at  transferred  c a l c u l a t o r by  a  the  into  were  Monroe  thr  communications  1880  data  program.  2 The  control  that  used  produce  set  was  could  plotted data  be  8).  data by  other  obtaining  the  a  slope,  v e l o c i t y and  control  calculated  the  partial  x  program scan  absorbance  vs.  radial  which set  was  of  not  data  9).  that  parameters was  which  molecular specific  programs mentioned,  the  , linear  with  to  in  the  a  weight, volume  8).  the After  small  sub-  after were  communications  turn  same  computed  (Figure  passed  distance  normally  Concurrently,  program  to  registers  plotting routine,  regression  statistical  vs. the  reserved  by  ln A from  output,  a  angular  the  a  ln  (Figure  that  of  in  the  obtained  of  This  to  set  routine  All  input  accessed  used  and  passed  values  stored  the  were  slope  of  (Figure  printed, that  then  some b a s i c  a  squared  was  the  supplied. program,  2 In  A  vs.  molecular  x  weight  regression  p l o t t i n g and  c a l c u l a t i o n subroutine,  were  the  present  in  F i g u r e 8.  Input and output o f the d a t a c o n v e r s i o n program and l i n e a r r e g r e s s i o n p l u s p l o t t i n g program.  Data c o n v e r s i o n program Input:  A - number o f data p o i n t s - f i r s t good absorbance v a l u e - baseline correction - baseline - p o s i t i o n o f f i r s t good v a l u e  Output: B - x 2 and l n A p a i r s  (not normally p r i n t e d ) 1  L i n e a r r e g r e s s i o n p l u s p l o t t i n g program 2 Input:  C - maximum and minimum f o r x - maximum and minimum f o r l n A - number o f d a t a p o i n t s t o be r e g r e s s e d - l o c a t i o n o f f i r s t data p a i r i n memory 2  Output: D - mean o f x - mean o f l n A - slope -  intercept  - standard d e v i a t i o n - correlation  coefficient  C a l c u l a t i o n o f m o l e c u l a r weight Input:  E - (1 -  X  vp)  x 10  6  Output: F - m o l e c u l a r weight Ln A values r e p r e s e n t the n a t u r a l l o g o f the v o l t a g e produced by the scanner, r a t h e r than true absorbance.  39.  2 3 «0 0 0 0 0 0 0 1 7 7- 0 0 0 0 0 0 0  I  -0000000 3 « 4 2 . < ooooooo 7 .15 5 8 0 0 0 5 0 .9336153 3 •0726933  A  I J B  —'  5 0 •7722172  51  3 .0252910  4  5 0 •5562793  . 0 0 0 0 0 0 0 7 - 0 0 0 0 0 0 0  3 • 1  2 •9123506  1 .  5 0 •3408016  1^  2 .3154087 5 0 •12 5 7 8 4 0 2 •7536607  0 0 0 0 0 0  7 0 0 0 0 0 0  • ••••••••• 1 7 . 0  0  0 0 0 0 0  9 2 * 0  0 0 0  0  00  4 9 • 9 1 12 2 6 6 2 .6390573 4  4 9 « 1 6 8 7 8 9 1  A  2 . 3 7 6 0 1 8 6  A  0 • 3 8 9 8  A  9 .6971294 2 «59 5 25 4 7  4 9 <4834924  37 3  1 6 * 7 9 1 8 0 9 4  A  2- 4 8 4 9 0 6 6 4 9 • 2 7 0 3 15 5 2 . 4 15 9 13 7  0 - 0 1 5 2 9 5 9  A  0•  A  9 9 9 2 0 0 4  4 9 «0 5 7 5 9 8 9 .-I  4  3 3 2 14 3 8  S '8  4 5 3 4 2  4  2 « 2 5 12 9 17 6  2• 4  3 3 5 461 3 32 30  0 * 2 5 1 0 0 0 0  6• 4 2 2 2 0 9 9  1 . 5 7 7 8 2 1 0  1 6  2 « 1 16 2 5 5 5 4  S «2  1 13 34 0  F  2 • 0 14 9 0 3 0. 4  3 . t  •  0 9 18 2 3 15 2 14  0 0 9  4 7• 7 9 0 9 6 2 6  l •8 2 4 54 92 4 7• 5 8 14 6 7 2 1 • 7 4 0 4 6 61  !  1  i j  i  t !  - 4-7,9. 9 6 * 1 8 5 9 3  40.  the  calculator  entered  the  and  the operator passed  i n p u t f o r each scan.  t h e c e n t r i f u g a t i o n data  less  l o n g and  i n a b o u t one  complicated  susceptible to errors.  Figure  9 was  typical  of  The  molecular weights  Table  I.  These r e s u l t s  weight of  this  system,  hour, making  calculation  The  plot  calculated  showed no  c o n d i t i o n s , and  a  simple  illustrated  the ones o b t a i n e d  run.  under the e x i s t i n g  Using  and  f o r a l l nine concentrations of  o v a l b u m i n were p r o c e s s e d previously  control  f o r the  and in ovalbumin  are presented  in  c o n c e n t r a t i o n dependency the average  molecular  4 5 , 3 0 0 d a l t o n s compared w e l l w i t h t h e v a l u e s i n  the  literature  (49).  Runs w i t h  and  conalbumin produced  similar  r e s p e c t i v e molecular weights  lysozyme, t r y p s i n results, with  being  inhibitor  their  14,400, 16,300  and  78,000 d a l t o n s .  Heterogeneous  Systems  Pseudo-ideal r o u t i n e l y by  h e t e r o g e n e o u s s y s t e m s c o u l d a l s o be  the data p r o c e s s i n g system.  heterogeneous systems, the m o l e c u l a r the  solution  In the case  weight v a r i e d  column o f t h e c e n t r i f u g e c e l l  handled  and  a  of  along non-  2 linear  f u n c t i o n was  obtained f o r the  l n A vs. x  In order t o evaluate the m o l e c u l a r weights  of these  2 the d e r i v a t i v e ,  d i n A/d(x  ) , had  t o be  plot.  evaluated.  systems,  2.9  3  51 radial distance squared (x ) Figure 9.  Plot of l n A vs.x for ovalbumin showing the least squares f i t to the data. 2  • J-  Table I.  M o l e c u l a r Weights Obtained f o r Ovalbumin*  Absorbance  M o l e c u l a r Weight (daltons)  0.20  45,959  0. 2 5  44,474  0.30  47,592  0. 3 5  44,485  0.40  44,169  0.45  47,996  0.50  46,498  0.55  44,815  0.60  45,638  * - @ 12,000  rpm  - v = .749*"  Average m o l e c u l a r weight - 4 5 , 3 0 0 Standard d e v i a t i o n - - 1 , 2 0 0  P a r t i a l s p e c i f i c volumes were o b t a i n e d from the Handbook of Biochemistry ( 4 9 ) .  T h i s c a p a b i l i t y was provided  by the m o d i f i c a t i o n o f a  m u l t i p l e r e g r e s s i o n program t h a t allowed automatic  fitting  2 of l n A v s . x  data t o any d e s i r e d p o l y n o m i a l .  From the  d e r i v e d p o l y n o m i a l c o e f f i c i e n t s , the b e s t f i t f u n c t i o n and  i t s d e r i v a t i v e s c o u l d be generated and p l o t t e d .  The  s a l i e n t f e a t u r e s of t h i s c a l c u l a t i o n can be i l l u s t r a t e d u s i n g experimental data obtained an enzyme p r e s e n t  from running c a t a l a s e ,  i n s o l u t i o n i n both i t s monomer and  tetramer forms. C a t a l a s e was d i s s o l v e d i n 0.05 M phosphate b u f f e r , pH and  c e n t r i f u g e d a t 8,300 rpm u n t i l e q u i l i b r i u m was reached.  The  c e n t r i f u g e data were i n i t i a l l y processed as d e s c r i b e d 2 f o r homogeneous systems. Since the p l o t o f l n A v s . x  data showed marked c o n c a v i t y values  were analysed  to the x a x i s , the s t o r e d  using a modified  multiple  regression  program that allowed automatic f i t t i n g o f l n A v s . x data by any d e s i r e d p o l y n o m i a l . In g e n e r a l , a q u a d r a t i c f u n c t i o n was s u i t a b l e .  Figure  10 i l l u s t r a t e s the p l o t o f  2 the l n A v s . x quadratic  data  ( c i r c l e s ) and the l e a s t squares  f i t ( s o l i d l i n e ) obtained  for catalase.  Figures  11 and 12 i l l u s t r a t e the i n p u t and output from the m u l t i p l e regression c a l c u l a t i o n . can be r e a d i l y obtained  The d e r i v a t i v e o f the f u n c t i o n from the back c a l c u l a t i o n f e a t u r e  o f the program, u s i n g the r e g r e s s i o n c o e f f i c i e n t s from the  relation:  Input and output of the m u l t i p l e r e g r e s s i o n program.  F i g u r e 11.  Input:  A - x  and l n p a i r s t o be r e g r e s s e d  B - number of data p a i r s t o be r e g r e s s e d C - number o f transforms D - t r a n s f o r m order Output: E - means F - standard d e v i a t i o n s G - correlation  coefficient  H - scandaru e r r o r of estimate I - F - ratio J - first  coefficient  - standard d e v i a t i o n o f the c o e f f i c i e n t -  t-test  46.  5 1 4 7 7 3  1_  5 0 0 0 00 5 0  1 5•0 10 0* 0 5 1 *2 6» 2 5 0 •9 5 •8 5 0 •6 5 •6 5 0•4  4 9 2,4 1 2 5  E  L  0 0 2 2 5 6 8 3 1  !  j i  5 • 9. 5 0• 1 4 5«2 4 4 9 •8 8 5 •0 8 4 9•6 1 4 •9 9 4 9* 3 5 4 *7 9 3  r L  * 4 8 5 7 1 4 2 6 4 9 •9 7 7 5 8 5 1,3 5 1 • 3 9 9 8 . 0 0 5 7 1 4 2 8  1 • 108 8 2 2 7 5 10 9 * 7 6 2 9 5 6 5  F  8,1 5 2 • U 0 6 5 6 0 * 5 9 5 3 3 3 3 7  G  —  0 * 9 9 7 8 1 6 3 0  H  --  0 * 0 3 1 7 2 2 5 8  M /  SO so SD SO  .  0 2 0 J 0 4  DO. e  3 • 1 0 . I  4 9• 8 4 •6 9 4 8 •8 2  "~  t  o  df o df /  .1,5 2 3 • 1 2 9 5 7 0  F  0  ;  4» 5 9 4 8•5 6 4» 4 9 4 8•3 0 4 •4 3  i  ;  -  1 74 *9 3 5 7 3 6 2 6 5 . 3 3 7 7 2 8 6 2 2 * 6 7 7 4 0 7 6 7  A  - 3 * 6 4 9 3 6 0 0 6  A  s  t  1.32041285 - 2 * 7 6 3 8 1 7 4 3  . 4 3 .0 3 4 •3 8 4 7 •7 7 4 •3 0 0 0 G 00 0 -  r L  J  1 •  0 *02 54 15 81 B  0 . 0 0 8 8 9 2 6 3 2 . 8 5 8 0 7 3 7 2  3« 0 0 0 0 0 0 0 0 —  C  12 3 - -  D  0• 2,7 9 5 * 1 7 2 3 0 6  1,0 7 7 . 4 4 0 8 1 3  S  :  t;  A s  F i g u r e 12.  Input and output o f t h e m u l t i p l e r e g r e s s i o n back c a l c u l a t i o n .  Normal back Input:  calculation  A - number of data  pairs  - number o f transforms - t r a n s f o r m order B - c o e f f i c i e n t s d e r i v e d from t h e best f i t 2 Output: C - b e s t f i t x and l n A v a l u e s Derivative  calculation - as above except t h a t the c o e f f i c i e n t s were changed  1 4  14*0 0 000000  2  3.00000000  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1.ooocoooo 2 . 0 0 0 0 0 0 0 0 3.0 0 0 0 0 0 0 0 •2,7 9 5 • 1 7 2 3 0 6 174.9 3 57362 64938006 0 2 5 4 15 8 1  22000000 18 12 4 4 0 3 95000000 89498457 6 8 0 0 0 0 00 6 4 30 5 0 31 4 10 0 0 0 0 0 42243966 1 4 0 0 0 0 0 0 2 3 0 15 10 9 88000000 0 6 8 9 5 0 5 2 610 0 0 0 0 0 92352832 35000000 3 0 18 8 9 6 9 08000000 6 9 17 6 8 19  ^ _J  ~~j I  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1 7 -7 0  •9357362 2 9 8 7 6 0 12 0 7 6 2 4 7 4 3  5 1 1  22000000  5 0 0  95000000 99480224  5 0 0  6 8 0 0 0 0 0 0 8 7 3 2 3 0 3 7  5 0  4 10 0 0 0 0 0 7 6277537  B  _J  9 5 0 0  12 7 4 9 0 9 8  14 0 0 0 0 0 0 6 6 3 4 3 7 2 5  4 9 0  8800000 0  4 9 0  6 10 0 0 0 0 0  4 9 0  35000000  4 9 0  5 7-8 2 8 5 2  1  5007 6 911  4366 3 086 08000000 3 8093677  y  y  1  =  a + bx + cx  + dx  =  b + 2cx + 3dx^  Eq.  13  Eq.  14  T h e r e f o r e , r e - e n t r y o f the c o e f f i c i e n t s a c c o r d i n g t o E q u a t i o n 14 produces the d e r i v a t i v e , and hence the p o i n t m o l e c u l a r weights  o f the s o l u t e a t e q u i l i b r i u m .  m o l e c u l a r weights  o b t a i n e d i n the above manner are  presented i n Table I I .  The m o l e c u l a r weights  The  i n the  r e g i o n o f the meniscus and those a t the c e l l bottom agreed w e l l w i t h the m o l e c u l a r weights  of 57,500 and  232,000 d a l t o n s f o r the monomer and tetramer  respectively  (49).  DISCUSSION  As a method f o r the d e t e r m i n a t i o n of m o l e c u l a r weights  of p r o t e i n s , the p r e p a r a t i v e UV  scanner had  a  d i s t i n c t advantage over o t h e r o p t i c a l systems, as i t allowed v e r y low c o n c e n t r a t i o n s to be a n a l y z e d .  The  low c o n c e n t r a t i o n s , i n g e n e r a l , o b v i a t e d the problems a s s o c i a t e d w i t h n o n i d e a l i t y , and t h i s was the case i n a l l the p r o t e i n s examined. the o p t i c a l system, however, was  found t o be  A drawback t o  i t s sensitivity  to  50.  Table  II.  x  Molecular  Weights  (cm )  Obtained  f o r Catalase*  Molecular Weight (daltons)  51.22  269,301_  50.95  237,627  50.68  208,581  50.41  182,186  50.14  158,466  49.88  138,114  49.61  119,602  49.35  104,290  49.08  90,985  4 8 . 82  80,690  48.56  72,831  48.30  67,456  48.03  64,494  47.77  64,136  *  - @ 8,300 -  v =  .730  rpm  UV  absorbing  contaminants.  The  correct  determination  of the b a s e l i n e a f t e r d e p l e t i o n of the sample a t end  of a run was  essential for obtaining  m o l e c u l a r weights.  presence o f UV derived  determination  absorbing  (46)  c o u l d be  noted  that  caused by  the  low m o l e c u l a r weight components  from d i a l y s i s t u b i n g .  r e f r a c t i v e index  reasonable  Schachman and E d e l s t e i n  e r r o r s i n the b a s e l i n e  the  O p t i c a l systems u s i n g  ( s c h l i e r e n and  r a l e i g h ) are  not  s u f f i c i e n t l y s e n s i t i v e t o s i m i l a r contaminants to p i c k up t h i s c o n t r i b u t i o n . d e p l e t i o n t r a c e s had air-buffer baseline. and  any  Another problem was absorbance v a l u e s The  bulletin  below t h a t o f  reason f o r t h i s was  runs w i t h t h i s anomaly were  Recently,  t h a t some  unknown  discarded.  Beckman Instruments p u b l i s h e d  ( 7 ) t h a t compared the p r e p a r a t i v e  to the Model E scanner.  UV  a technical scanner  Many o f the f a c t o r s t h a t  a f f e c t the accuracy of the scanner r e s u l t s , such as precesion,  optical linearity,  the  could rotor  scanner r e p r o d u c i b i l i t y and  performance o f the e l e c t r o n i c s of the u n i t , were i n v e s t i g a t e d . In the f i n a l a n a l y s i s , the p r e p a r a t i v e  UV  scanner  deemed f o r a l l p r a c t i c a l purposes t o be e q u i v a l e n t Model E scanner, i f the d r i v e and up t o standard.  to  the  o p t i c s were shown to  A f u r t h e r note was  made concerning  problem o f o b t a i n i n g the zero b a s e l i n e v a l u e by the u n c e r t a i n t i e s o f which c o u l d  was  be  the  depletion,  lead to v a r i a t i o n s i n  molecular this  weight  problem  scanner, In  b u t was  general,  concurred  The  data  well,  programs, and  obtained system  was  the loading  o f c o n t r o l from  once  and e r r o r prone the procedure  to function readily.  weights, f o r became of the input  one program  was  required  a  routine  appropriate parameters  to  another.  initially,  c a l c u l a t i o n s were  was  also.  work.  selected data  systems,  UV  bulletin  proven  molecular  effort  noted,  scanner  i n this  t h e p r o v i s i o n o f some b a s i c  a programming  E  i n t h e Beckman  into  (7)  to the preparation  manually  requiring only  consuming  performed  of data  Chervenka  i n the Model  and heterogeneous  the passing  Although time  acquisition  reproducing  homogeneous procedure,  prevalent  the results  processing  As  not r e s t r i c t e d  the conclusions  with  The very  was  estimations.  established.  the  easily  PART I I .  MOLECULAR WEIGHT DISTRIBUTIONS  THEORY  As m e n t i o n e d  previously,  sedimentation  equilibrium  u l t r a c e n t r i f u g a t i o n has t h e p o t e n t i a l f o r p r o v i d i n g large the  amount o f i n f o r m a t i o n  system.  centrifuge,described solutes  Early Rinde  i n t h e development (44) d e r i v e d  comprising  of the u l t r a -  an i n t e g r a l r e l a t i o n t h a t  the complete m o l e c u l a r weight d i s t r i b u t i o n o f  when t h e s y s t e m was a t e q u i l i b r i u m .  e q u a t i o n c a n be d e r i v e d cells  about t h e s o l u t e s  a  f o r sector  shaped  The Rinde centrifuge  by s t a r t i n g w i t h t h e m o l e c u l a r w e i g h t r e l a t i o n  (Equation pertain  12) o b t a i n e d  only  t o i d e a l systems  h a v e t h e same p a r t i a l coefficient. stated  i n Part  I. that  specific  are incompressible  and  volume a n d e x t i n c t i o n  F o r a multicomponent  l i m i t a t i o n s , Equation  The f o l l o w i n g e q u a t i o n s  system, w i t h i n t h e  12 c a n be e x p r e s s e d a s  follows: 2  din c .  E q . 15 2RT  or dc.  M. (1 - v . p ) oa c •  2RT  E q . 16  Since  partial  are  assumed  can  be  specific  t o be  volumes  t h e same  and  extinction  f o r a i l components,  coefficients a new  term  defined:  " ^  ( 1  p ) f a ) 2  Eq.  17  Eq.  18  2RT  Solving  Equation  16  for i  dc = d(r  The  AcM  =  A  components  ^ /^  c i  )  M  1  following  r,  definitions  -  r  -  r  were  results in:  i  1  i n i t i a t e d by  Fujita  (19):  Eq. m  where: b  -  i s the c e l l  m  -  i s the  bottom  solution  meniscus  19  X  =  (1  - vp)co (r? t> 2  Using Equations  din  c. —  - r )/2RT m 2  19 and 20, Equation  =  Eq.  20  16 can be converted  to  Eq.  21  -AM.  dC I n t e g r a t i n g t h i s equation between the c e l l bottom (.£ = 0) and some r a d i a l p o s i t i o n £ :  In  c (C)/c (5=0) I  i  =  -XM.5 l  Eq.  22  Eq.  23  or:  c (?) i  Equation  =  c U=6) ±  exp(-XM 5) I  22 has been d e r i v e d w i t h o u t any r e f e r e n c e t o the  shape o f the c e l l or the i n i t i a l c o n c e n t r a t i o n o f the sampl By u s i n g the law o f the c o n s e r v a t i o n o f mass, the i n i t i a l c o n c e n t r a t i o n o f each component shaped c e l l can be d e s c r i b e d .  of the sample o f a s e c t o r Here 6 i s the s e c t o r  and h the l e n g t h of the s o l u t i o n column.-  angle  56.  r  Gh  /  b c,d(r )  =  d£  c  2  6  h  o i  c  (  r  m  ~  r  b  }  E  q  "  2  4  or  =  .  Eq.  oi  25  0  Where The  C ^ q  total  summing  i s the i n i t i a l  concentration  concentration of the c e l l  a l l the i n i t i a l  of the i  can be  component.  expressed  concentrations o f each  of  by  q  components:  q 'o  i  ^  °  o  i  i=l  By  c a n be  Equation  -  Equation  o  (V  Equation  25,  =  26  the  .  „ E  M.c . c,  into  = 1  Substituting  23  Eq.  obtained:  .XM.C c.(5=0)  cd£  0  substituting  following  I  1  -  q  ' on '  2  7  exp(-XM ) i  27  into  Equation  exp(-XM.e) exp(-XM ) i  23  gives:  E  q  _  2  g  Summing a l l the components i n the system and the t o t a l t o the i n i t i a l  (5  )  = fifi  _  c o n c e n t r a t i o n produces:  V>  o  AM.  "', ,* 1=1  Substituting  f  i  = c  o i  relating  /c  exp(-XM.g)  1 - exp(-AM.) exp(-AM ^ 1  f o r the weight  c . o  E g >  2 9  c o  f r a c t i o n of  component i , the equation becomes:  XM  c(£)  ±  exp(-AM ^) I  E c f  c  o  • . , i=l  1 - exp ^  I f the molecular weights  (-AM.  1  )  I - 30  i  of the components o f the  system  r e p r e s e n t a continuous d i s t r i b u t i o n , then E q u a t i o n 30  can  be w r i t t e n i n the g e n e r a l form as f o l l o w s : CO  c(5)  / I  = C  I  Q  AM  exp(-AMO f(M)dM  1 - exp  JJ  (-AM)  0 where f ( M ) r e p r e s e n t s a d i f f e r e n t i a l m o l e c u l a r d i s t r i b u t i o n of the sample components.  weight  T h i s i s the  Rinde equation upon which a l l the methods f o r attempting  -  58.  to  o b t a i n a c o m p l e t e MWD  restricted The  i n i t s application  systems.  b y S o u c e k and Adams  t o i n c l u d e some n o n i d e a l s y s t e m s b y u s i n g t h e l i g h t  scattering v i r i a l ^ ideality.  Wan  coefficient  Although useful  cells  Donnelly  appeared  the Rinde equation  almost  much more v e r s a t i l e , multimodal  Programming  recently,  when  s i m u l t a n e o u s l y , one b y (14,47).  Both  solutions  the problem o f negative  i n a l l previous  submitted  solutions.  by S c h o l t e proved  as i t was c a p a b l e  t o be  of analyzing  - The S o l u t i o n o f S c h o l t e  solution  some o f t h e e a r l i e r  the problem.  involved a similar  attempts  equations, but d i f f e r e d  Equation  i n 1928,  distributions.  Scholte's  solve  allowed  i t s publication  circumvented  t h a t had appeared  However, t h e s o l u t i o n  Linear  after  and t h e o t h e r by S c h o l t e  frequencies  that  i n o b t a i n i n g MWDs.  s o l u t i o n d i d n o t appear u n t i l  solutions  f o r non-  t h e R i n d e e q u a t i o n was r e c o g n i z e d t o b e a  development s h o r t l y  a practical  (B, ) t o c o r r e c t Is  (59) a l s o d e r i v e d e q u a t i o n s  the use o f Yphantis  to  and w h i c h i s  to pseudo-ideal  Rinde e q u a t i o n has been extended  (50)  two  have b e e n b a s e d  i n i t s use o f  simultaneous  i n i t s interpretation  Scholte f i r s t  31 a s f o l l o w s :  approach t o  o f how t o  o f a l l expressed  59.  q Eq. c  32  i=l  o  where the s u b s c r i p t j i n f e r s t h a t X i s a v a r i a b l e . Equation  3 2 d e s c r i b e s the c o n c e n t r a t i o n a t any  position  t, as the sum  present  of a l l the molecular  radial  weights  i n the system at t h a t r a d i a l p o s i t i o n .  c o n c e n t r a t i o n at t h a t p o i n t c(£)  The  i s r e l a t e d t o tl\i  frequency  of the presence of a l l the c o n t r i b u t i n g  molecular  weights.  two  As the equation  unknowns, the molecular  stands  there  are  weights c o n t r i b u t i n g t o  the c o n c e n t r a t i o n a t any d e s i r e d r a d i a l p o s i t i o n and the frequency  of t h e i r presence.  parameters can be o b t a i n e d S c h o l t e reduced the two  A l l the  other  from experimental  unknowns t o one by  data. arbitrarily  a s s i g n i n g an e x p o n e n t i a l l y spaced molecular  weight  s e r i e s c o v e r i n g the whole spectrum o f molecular present  i n the sample, and  frequency  v a l u e s by  s o l v e d the e q u a t i o n  l i n e a r programming.  weights f o r the  A detailed:  summary of the c a l c u l a t i o n procedure f o l l o w s : S c h o l t e f i r s t of a l l rewrote Equation  f. I  o  32  as;  Eq.  33  60.  where X .M. -2-Ji  exp (-X .M. E,) ___ 1 - exp(-AjM )  =  K 1 3  E  q  >  34  i  6j i s a s l a c k v a r i a b l e i n c l u d e d to a l l o w f o r error.  By expanding Equation  equations  33,  a s e t of  i s formed t h a t accounts  experimental  simultaneous  f o r the changing  c o n c e n t r a t i o n i n r e l a t i o n to r a d i a l d i s t a n c e . C  ^ l  )  /  c  o  c(C )/c 2  Q  Vll  =  =  f K x  +  1 2  f  2 21 K  + f K 2  +  2 2  f  3 31 K  +  +. f K 3  +  3 2  6  1  + •-• +  6  2  Eq.  c(E )/c n o  =  f,K, + f K+ f-K+ ••• + 1 ln 2 2n 3 3n  35  6. j  A measurement of the c o n c e n t r a t i o n f o r a g i v e n lambda (a f u n c t i o n of r o t o r speed) at any r a d i a l p o s i t i o n would p r o v i d e one  c(£)/c  value.  T h e r e f o r e , the a n a l y s i s o f  u l t r a c e n t r i f u g a l data a t v a r i o u s speeds, and the measurement o f the c o n c e n t r a t i o n a t a number o f r a d i a l p o s i t i o n s p r o v i d e a l l the data r e q u i r e d except weights.  The  f o r the  molecular  p r o v i s i o n of a m o l e c u l a r weight s e r i e s would  a l l o w the c a l c u l a t i o n of the K ^ j  term i n each  equation,  leaving of  the frequency  overdetermined  Equation  avoid negative  of  positive  to  be o b t a i n e d  term of  6.. 3  complete  obtained  the  equations  solved by  This  constraints.  a n d a t t h e same  range  solution  time  would  weights  present  another  express  m a n n e r was  would  s e t o f simultaneous set of molecular  equal  that  1/4 a  factor  of 2 '  was  repeated  a total  set  of solutions  the i n i t i a l of four  that  weight  to provide  Stated  mathematically:  E i f  =  one.  again,  The  by  solution used solving  but using  were s h i f t e d  over  by  , This  providing  c o u l d be p l o t t e d  a complete  covered  Scholte  series.  times,  to  t h e sum o f t h e  . from  chosen  t o advantage  equations  solution  the error  weights  not unique.  weights  the use  the frequency  arbitrarily  property of the calculation same  minimized  i n the system,  by  programming.  a realistic  I f the molecular  of the solution  i n this  linear  allowed  The s e t  illustrated  frequencies, Scholte proposed  the system.  frequencies  this  be  of the molecular  represent the  then  The f i n a l  each  as t h e o n l y unknown.  simultaneous  35 could  To  value  a  procedure larger  against molecular  molecular weight  for  any one  series  for  a l l series  distribution.  Eq.  36  Eq.  37  1  4  therefore: X) .j/ f  4  =  1  for a l l series  Eq.  38  Through experience S c h o l t e found t h a t an e x p o n e n t i a l m o l e c u l a r weight s e r i e s w i t h a s p a c i n g o f m u l t i p l e s o f two between m o l e c u l a r weights  p r o v i d e d the b e s t  In summation, S c h o l t e ' s procedure complete MWD 1.  solution.  f o r o b t a i n i n g the  involved:  A r b i t r a r i l y s e l e c t i n g a m o l e c u l a r weight  c o v e r i n g the range of the m o l e c u l a r weights  series  present i n  the sample. 2.  C a l c u l a t i n g the K^j terms u s i n g the m o l e c u l a r  weight s e r i e s and equating them t o the 3.  S o l v i n g the simultaneous  programming w i t h p o s i t i v e 4. by  C  (  ^  )  /  C  q  equations by  values. linear  constraints.  I n c r e a s i n g the o r i g i n a l m o l e c u l a r weight  series  2 ' . 1  4  5.  Repeating  steps 2-4  t h r e e more times.  6.  P l o t t i n g the f r e q u e n c i e s o b t a i n e d a g a i n s t  m o l e c u l a r weight t o produce a complete  MWD.  M u l t i p l e Regression as an A l t e r n a t i v e S o l u t i o n  Magar (30) reviewed complete MWDs  a l l the major methods of o b t a i n i n g  v i a equilibrium ultracentrifugation.  r e f e r e n c e t o the method of S c h o l t e , Magar suggested  With the  use  of multiple  programming  regression  for obtaining  argument i n f a v o r availability errors,  state  the  function  i n t e r v a l s and c o r r e l a t i o n c o e f f i c i e n t s  data.  f i t o f the  T h i s would enable the  degree o f p r o b a b i l i t y with approximated  these considerations  is  parameters such as standard  e s t a b l i s h the  were r e a s o n a b l e ,  the  f o r the  there  a problem avoided  into  Since  linear  frequencies  i nthelinear  p o s i t i v e constraints could  (least  Magar s u g g e s t e d t h e  he recommended t h e  frequency problem.  multiple regression to a rather  (37).  solution,  not  be b u i l t  use o f a norm  I f these curve  Use o f t h e  a p p r o a c h a s s u g g e s t e d b y Magar  lengthy  search  Therefore,  using the  occurred.  be  derived  f r o m Magar's d i s c u s s i o n was t h a t  basic  c o u l d be u s e f u l i n o b t a i n i n g  i npractice this  approach d i d not  could  a l t e r n a t i v e methods  failure  but  This  i n the  use o f other  if  regression  regression  t h a t had p o s i t i v e c o n s t r a i n t s ,  t h e method o f N e l d e r a n d Mead  lead  Although  s q u a r e s method) o r i n f i n i t y  methods f a i l e d ,  procedures  distribution  p r o g r a m m i n g method o f  negative  or  which the  use o f m u l t i p l e  t o overcome t h e  fitting  researcher t o  i s one m a j o r d e t e r r e n t .  regression,  constrained  distribution to  true d i s t r i b u t i o n .  t h e appearance o f negative  Scholte.  The major  o f m u l t i p l e r e g r e s s i o n was t h e  confidence  original  c o m p l e t e MWDs.  of statistical  that could help the  as an a l t e r n a t i v e t o l i n e a r  conclusion t o multiple  c o m p l e t e MWDs,  seem t o b e f e a s i b l e .  However, regression the  as Magar's c o n c e p t o f u s i n g  had m e r i t ,  an e n d e a v o r was made t o o v e r c o m e  l i m i t a t i o n s j u s t mentioned.  using  Magar s suggestion 1  in  only  f o r t h e MWD  no c h a n g e s i n t h e  a d v o c a t e d by  Scholte.  d i f f e r e n c e between t h e two a p p r o a c h e s was  multiple  minimized, the  Solving  required  concept o r approach o r i g i n a l l y The  multiple  regression rather  that  the squares o f the d e v i a t i o n s  than t h e a b s o l u t e  deviations.  were  Following  s u g g e s t i o n s made by Magar, t h e s o l u t i o n o f t h e MWD  calculation first  using multiple  a t t e m p t s were f o c u s e d  Scholte  i n hisoriginal  program  supplied  used.  However,  those obtained  regression  was t r i e d .  on t h e d a t a p r e s e n t e d b y  article.  A multiple  regression  by t h e Monroe c a l c u l a t o r company initial  The  was  r e s u l t s d i d n o t a t a l l resemble  by S c h o l t e .  The p r o b l e m v/as f o u n d t o be  t h e p r e s e n c e o f an i n t e r c e p t t e r m i n t h e m u l t i p l e regression  solution.  Since  i t was n o t p o s s i b l e t o  r e p r o g r a m t h e Monroe c a l c u l a t o r t o p r o v i d e went t h r o u g h t h e o r i g i n ,  an a r t i f i c i a l  a solution  method was  that  devised.  Removal o f t h e i n t e r c e p t was a c c o m p l i s h e d b y e n t e r i n g an extra  dependent v a r i a b l e and a c o r r e s p o n d i n g  set of  i n d e p e n d e n t v a r i a b l e s , a l l o f w h i c h were z e r o e s . the  final  m a t r i x t o be s o l v e d  would appear a s :  Therefore,  c(  ? 1  )/c  0  K  ll  K  21  K  31  c(  ? 2  )/c  o  K  12  K  22  K  32  K  ml  K  m2 Eq. 39  C  The  ^n 0  ) / c  o  K  ln  K  K  2n  3n 0  K  mn 0  a d d i t i o n o f t h i s n u l l s e t a t the end o f the s e r i e s o f  simultaneous  equations  to be s o l v e d e f f e c t i v e l y removed  the i n t e r c e p t term i n the f i n a l s o l u t i o n , by r e d u c i n g i t s v a l u e very c l o s e t o zero.  Upon the i n c l u s i o n o f the n u l l  s e t , the a n a l y s i s o f S c h o l t e ' s data allowed  the r e -  p r o d u c t i o n o f the d i s t r i b u t i o n shown on page 116 o f h i s paper (47). T h i s p o s i t i v e r e s u l t was, however, s h o r t l i v e d when other d i s t r i b u t i o n s were examined.  As soon  as a more r e s o l v e d d i s t r i b u t i o n was i n v e s t i g a t e d , u s i n g m u l t i p l e r e g r e s s i o n , the negative f r e q u e n c i e s noted by Magar (31) appeared. suggested  Since the a l t e r n a t i v e methods  by Magar were not r e a d i l y a v a i l a b l e ,  approach was taken. produced negative  another  The molecular weights t h a t had  f r e q u e n c i e s were removed, on the premise  t h a t , i n r e a l i t y , they were not p a r t o f the s o l u t i o n . The procedure can be i l l u s t r a t e d as f o l l o w s :  Assume t h a t f i v e m o l e c u l a r weights used t o f i t the MWD.  comprised  I f the t h i r d produced  a  series  a negative  frequency;  f  l l M  f  2 2  " 3 3  M  f  M  f  4 4 M  f  5 5  '  M  then the c a l c u l a t i o n would be repeated u s i n g the  :  1 1 M  f  2 2 M  f  4 4 M  f  5 5 M  E q  "  4 0  following:  •  E q  '  4  1  I f the removal o f the m o l e c u l a r weight a s s o c i a t e d w i t h the n e g a t i v e frequency produced  a set of only p o s i t i v e  f r e q u e n c i e s , the c a l c u l a t i o n would then be moved onto next s e r i e s .  the  T h i s s e r i e s would be s h i f t e d by a f a c t o r  1/4 of 2 '  from the f i r s t ,  I f t h i s was  an i n t e r v a l proposed  by S c h o l t e .  not the case, the n e g a t i v e m o l e c u l a r weight  would a g a i n be removed, u n t i l o n l y a p o s i t i v e was  produced.  The s o l u t i o n s  solution  o b t a i n e d by u s i n g t h i s  procedure were then compared t o those o b t a i n e d by S c h o l t e " s l i n e a r programming method.  Initial  comparisons i l l u s t r a t e d  a s t r i k i n g resemblance between the two i t was  difficult  without e r r o r s .  solutions.  to go through the complete T h e r e f o r e , i t was  However,  calculation  decided t o program and  automate a l l f a c e t s of t h i s c a l c u l a t i o n on the Monroe calculator,  t o a l l o w f o r e a s i e r assessment and  of t h i s approach.  1880  comparison  The r e s u l t i n g program r e q u i r e d 4,000  program steps and almost a l l 512 data r e g i s t e r s  available,  t a x i n g the c a p a b i l i t i e s of the c a l c u l a t o r t o t h e l i m i t . The program allowed the n e g a t i v e m o l e c u l a r weights t o be removed without r e q u i r i n g  the r e c a l c u l a t i o n of the  o r i g i n a l m a t r i x and removed much o f t h e tedium  of the  calculation. Using t h i s program, i t was found t h a t t h e removal of n e g a t i v e m o l e c u l a r weights  allowed r e p r o d u c t i o n o f  the r e s u l t s obtained by S c h o l t e ' s program, i n t h e simpler cases.  When two or t h r e e negative v a l u e s appeared  i n the  s o l u t i o n r a t h e r than one, the c h o i c e o f the o r d e r i n which the n e g a t i v e f r e q u e n c i e s were removed l e d t o different solutions.  Through t r i a l and e r r o r ,  found t h a t t h e removal o f the l a r g e s t n e g a t i v e  i t was frequency  produced  s i m i l a r r e s u l t s when compared w i t h S c h o l t e ' s  method.  Although t h i s program was much e a s i e r  than the o r i g i n a l , i t was s t i l l requiring  approximately  calculation.  to run  v e r y time consuming,  one hour o r more t o complete a  I t became apparent  t h a t the Monroe v e r s i o n  of the m u l t i p l e r e g r e s s i o n program was a b l e t o produce comparable d i s t r i b u t i o n s to those o b t a i n e d from S c h o l t e ' s l i n e a r programming approach.  Due t o the time consuming  nature of the c a l c u l a t i o n , a FORTRAN v e r s i o n was developed f o r use w i t h the U.B.C. IBM 37 0/168  computer, i n order t o  reduce t h e c a l c u l a t i o n time and t o allow more r o u t i n e t e s t i n g .  68,  During the i n v e s t i g a t i o n s w i t h the Monroe 1880 i t was  program,  observed t h a t the s t a t i s t i c a l parameters were very-  s e n s i t i v e t o the f i t of the chosen m o l e c u l a r weight s e r i e s , when the m o l e c u l a r weights were c o i n c i d e n t w i t h the maxima of the d i s t r i b u t i o n s .  T h i s l e d t o an  iterative  concept t h a t used the s t a t i s t i c a l parameters  from the  r e g r e s s i o n as a guide t o f i n d i n g the c o r r e c t m o l e c u l a r weights and c o n c e n t r a t i o n s of the components.  A further  expansion of the dummy v a r i a b l e concept, used i n removing the i n t e r c e p t from the o r i g i n a l m u l t i p l e r e g r e s s i o n program, was  developed t o smooth 'undefined' MWDs.  Both  of these concepts, o r i g i n a l l y noted i n the Monroe v e r s i o n , were i n c l u d e d discussed using  i n the FORTRAN program.  Some of the r e s u l t s  i n the next s e c t i o n s were o r i g i n a l l y o b t a i n e d :  the Monroe v e r s i o n , but c o u l d be more r e a d i l y  i l l u s t r a t e d i n the c o n t e x t of the completed  RESULTS AND  FORTRAN program.  DISCUSSION  Model Systems  In o r d e r to study and compare the l i n e a r programming and m u l t i p l e r e g r e s s i o n approaches,  and t o examine the  c a p a b i l i t i e s and l i m i t a t i o n s of each i n r e l a t i o n t o the study of p r o t e i n s , data were generated t o r e p r e s e n t e r r o r  f r e e models f o r study. model data was  One  approach used t o produce the  t h a t o f S c h o l t e (48), assuming a log. normal  d i s t r i b u t i o n , and the o t h e r was  a simple expansion  of the  Rinde e q u a t i o n t h a t d i d not assume a d i s t r i b u t i o n .  A.  Log n o r m a l ' d i s t r i b u t i o n method  In h i s study, S c h o l t e assumed t h a t a l o g normal d i s t r i b u t i o n d e s c r i b e d the m o l e c u l a r weights polymers.  The c h o i c e of the d i s t r i b u t i o n f u n c t i o n was  a r b i t r a r y , and was  of s y n t h e t i c  i n S c h o l t e ' s work the l o g normal f u n c t i o n  convenient, s i n c e the m o l e c u l a r weight s e r i e s he  to d e s c r i b e the systems was Using S c h o l t e ' s approach, generate  logarithmically  a program was  l o g normal d i s t r i b u t i o n s .  used  spaced.  w r i t t e n to  The b a s i c l o g normal  f u n c t i o n can be r e p r e s e n t e d as:  f (M)  Eq.  exp  42  where: S.D.  - i s the standard d e v i a t i o n of the  distribution  - i s the m o l e c u l a r weight a s s o c i a t e d w i t h the maximum frequency of the d i s t r i b u t i o n M  i s the v a r i a b l e m o l e c u l a r weight  f (M)  the frequency  obtained  70.  Equation to  42 represents  a unimodal d i s t r i b u t i o n .  produce a multimodal  distribution,  a series  In order o f these  f u n c t i o n s w i t h d i f f e r i n g maxima w o u l d be summed; f o r example, a t r i m o d a l d i s t r i b u t i o n following  i s represented  by t h e  relation: 1  f (M)  =  C  M  R-L exp -  In  +  S.D..  Eq.  M R  2  exp^-  M  ln  +  R  3  ln  exp S.D..  M,  S.D,  43  M'_,  where: R^_  a r e t h e p r o p o r t i o n s o f e a c h mode o f t h e dis-*:;'.  3  tribution S.D.^_  a r e the standard  3  M  l-3  a  r  e  t 3 i e  m  °l  e  c  u  l  a  r  d e v i a t i o n s o f e a c h mode weights a s s o c i a t e d with the  maximum f r e q u e n c y C  The  constant  standard  f o r v a r y i n g t h e sum o f t h e d i s t r i b u t i o n  above e q u a t i o n  generation  o f e a c h mode  o f f(M) a f t e r  was programmed t o a l l o w t h e t h e p r o p o r t i o n s , means, and  d e v i a t i o n s o f t h e components had b e e n d e f i n e d .  T h i s p r o g r a m was  made a s u b r o u t i n e  so t h a t t h e d i s t r i b u t i o n calculated.  of a plotting  c o u l d be p l o t t e d  The i n p u t , o u t p u t  and p l o t  program  a s i t was  of a three  71.  component system  a r e p r e s e n t e d i n F i g u r e s 13 and  14.  In o r d e r to c o n v e r t the d i s t r i b u t i o n data i n t o the d e s i r e d e x p e r i m e n t a l v a l u e s , a second program c a l c u l a t e d v a l u e s from the f o l l o w i n g expansion c(5)  c  o f the Rinde e q u a t i o n .  A.M .exp(-X.M £) - l - i ^ — — — f (M,) 1 - exp(-A.M ) j 1  =  0  1  o  1  X.M exp(-X.M O j n • j n 1 - exp(-X.M  ^ The  X M_ e x p ( - X .M ?) --—£ ^-JL 1 - exp(-X_.M„) j 2  +  speed,  )  f  (M  )  Eq.  c(5)/c  data.  and the m o l e c u l a r weights  The  Simple expansion  An a l t e r n a t i v e  and  f(M) the  i n p u t and output o f t h i s 16.  o f the Rinde e q u a t i o n  method to the l o g normal  the use of a simple expansion  Rather than u s i n g a d i s t r i b u t i o n discrete  44  j n  program i s i l l u s t r a t e d i n F i g u r e  was  M  r  o b t a i n e d from the above program entered t o generate  B.  (  p a r t i a l s p e c i f i c volume and r a d i a l d i s t a n c e  i n t e r v a l were chosen,  5 and  F  distribution  o f the Rinde e q u a t i o n . t o d e s c r i b e the system,  m o l e c u l a r weights were entered i n t o the Rinde  e q u a t i o n , a l o n g w i t h the r e l a t i v e r a t i o s o f the components, The Rinde e q u a t i o n was the f o l l o w i n g  function:  s o l v e d f o r c(£)/c  according to  )  F i g u r e 13.  Input:  Input and output of the l o g normal d i s t r i b u t i o n program.  A - plotting  instructions  B - m o l e c u l a r weight  o f component 1  - standard d e v i a t i o n o f component 1 - f r a c t i o n a l r a t i o o f component 1 C - same as B f o r component 2 D - same as B f o r component 3 E - more p l o t t i n g Output: F - l n m o l e c u l a r - f(M)  instructions weight  0 <• 0 0  0 • 0 0  i 0 0  2 • 0 0'  0 «• 0 0  0• 17  1 0  0 • 0 0  4«- 0 0  2 « 0 0  0 • 0 0 C) 0 5  0 • 0 0 A  0 • 5 0  4 • 17 0 • 0 1  0 • 0 0  o. 0 2 0 • c) 0 5  4 « 3 4  1 0 ' 0 1 0 • 0 1• 0 0 «- 1  0 • 2 0  0 0 0  4 « 5 1  0  0 •- 1 0  2 5,0 0 0 '• 0 0 0 «. 1 0 0 <« 3 3  nB J  1 0  0 » 3 3 3 2 0,0 0 0 . 0 0 0 • 1 0 0 » 3 3 0 • 72  4' • 0 0  0 •» 0 2 4 -. 8 6 0 • 22  8 0,0 0 0 <- 0 0 0  4 •• 6 9  j n  j  C 5 - 0 3 0 » 0 8  D  5.• 2 1 04 0 • 3(  5 • 38 0 • 11  74.  0.3  0.2 Y  0.1  0.0  1024 M x 10 Figure 14.  Sendlcgarithmic plot of the output, f (M) vs. M, from the log normal distribution program for a three component system of 25,000-80,000-320,000 daltons (1:1:1 ratio).  1.0  0.6  0.2  224 M x 10  Figure 15. Semilogarithmic plot of f(M) vs. M for a 25,00080,000 dalton mixture (1:1 ratio). Multispeed data; interval =2.0.  F i g u r e 16.  Input:  Input and output o f the K v s . c ( ? ) / c program f o r the l o g normal d i s t r i b u t i o n . o  A - p a r t i a l s p e c i f i c volume - temperature °C - distance  from c e l l bottom t o the meniscus  - number of e n t r i e s B - l n m o l e c u l a r weight - frequency C - solution - cell  meniscus  bottom  - starting position - delta x - number of output v a l u e s - rpm Output: D - c h a r t  position  - K  ~ cU)/c_ E - X  76.  1 0*700  |  6 2*3333  ^ A  8  2 2 0 . • 00  5*0000  3 8 5  J  6 2 * 3 3 3 3 1 0  4 •  5 *6 6  66  11 4 * 0000 4*1700 1 4,7 9 1 « 0*0100  1  1 2 1 2,0 0 0 •  B 6 2*3  5 •  0 0 0 0 4.3700  14 5 9  2 3,4 4 2 •  6•  1  D  J  6 7*9 0*2000  7 5 5 3 2 0 6,8  4*5100  7 3 •6  3 2,3 5 9 •  5 0 71 2 9 6 9  0• 10 0 0  7 4*6900 4 6,9 7 7 .  0 0  9*3  2 5 5 3 4 3 21  6 • 0 •0 2 0 0 5 * 4 4 2 3 0 0 0 0 0  -05  77.  cU)  A .ML exp(-X.M £) _J ± ! J i f 1 - exp(-X.M ) J x  =  c  1  O  X.M 1  exp(-X M J-^ - exp(-X .M.)  3  This  equation  represented  up  E q . 45  3  T h i s approach probably  of d i s c r e t e molecular  for this  distribution  represents the  weights r a t h e r than  i n s y n t h e t i c polymers.  t h e programs used  The i n p u t a n d  t o generate  of the c a p a b i l i t i e s i n t h i s work.  having the  i n F i g u r e 17.  a p p r o a c h e s were u s e d  systems f o r t h e study  ratio  s i n c e p r o t e i n s a r e made  program a r e p r e s e n t e d  Both o f these  was f a v o u r e d  /•  b y t h r e e components i n some f r a c t i o n a l  wide range p r e s e n t  of  ]  f  of proteins i n solution,  output  2  describes a discontinuous  to each other! state  X.M exp(-X.M O _ J _ J—± f 1 - exp(-X.M )  +  and l i m i t a t i o n s  The R i n d e  somewhat due t o i t s s i m p l e r  model  equation  and f a s t e r  operation.  Molecular  Weight D i s t r i b u t i o n  Program U s i n g  Multiple  Regression  At  this  point, i t i s pertinent to describe  t h e b a s i c s o f t h e FORTRAN v e r s i o n o f t h e MWD  some o f  calculation,  Input and output of t h e Rinde e q u a t i o n program.  F i g u r e 17.  Input:  A - cell  bottom  - meniscus - number o f data p o i n t s d e s i r e d B - A - number o f components Output: C - £  -  c(K)/c  Q  7 • 2 0 000000 6 • 80 00000 0 4 •0 0 0 0 0 0 0 0 0 • 1 0 0 00C 00 5 • 60000000 0 • 00005079 2 • 00 000000 2 5,00 0 « 0 0 0 0 0 8 0,00 0 • 0 0 0 0 0 0 • 5 0 000000 0 • 50000000 1 1 0• 0 0 0 0 0 0 0 0 •0 0 0 0 0 0 0 0 2 • 9 5 0 0 1 89 4 o• 2553571 4 1 . 3 7 0 80 8 5 3 0 •5 0 7 1 4 2 S 5 0 • 72 7002 87 o. 7 5 5 3 5 7 1 4 0 •4 3 4 3 8 5 6 8  80.  summarize  some o f t h e m a i n s u b r o u t i n e s and  t h e o u t p u t p r o v i d e d by  this  program.  The  program  w r i t t e n w i t h the  comparison  Scholte s  programming p r o g r a m i n m i n d .  linear  1  the  i n p u t formats  c o u l d be either  readily  were made i d e n t i c a l  was  t o those  of  Therefore,  so t h a t  any  data  interchanged to allow c a l c u l a t i o n  by  program.  The  p r o g r a m was  routines  that w i l l  composed o f a number o f m a j o r  be  A two  r e q u i r e d f o r the  o f s o l u t e s was  molecular weight. weights  matrix  dimensional matrix m x n c o n s i s t i n g  molecular weights  (m)  calculation  and  generated  The  matrix  The  of  of overdetermined  required  the  starting  molecular  variables.  simultaneous 34.  input data  f o r the K^j  The set.  e q u a t i o n s was £ and The  c(£)/c  i n t h i s manner was  matrix.  The  the complete  set  calculated o  v a l u e s were  molecular  t e r m were drawn f r o m  c a l c u l a t e d molecular weight generated  the  values r e q u i r e d t o p r o v i d e the  to Equation  p r o v i d e d by  to cover  a single  number o f  of a l l the  t h e number o f c y c l e s r e q u i r e d f o r t h e  Regression  according  series  from  interval,  (p) were i n p u t  matrix  sub-  b r i e f l y described:  M o l e c u l a r Weight  range  of the r e s u l t s  illustrate  weights  the p r e v i o u s l y matrix matrix required  81.  f o r a l l p c y c l e s of the c a l c u l a t i o n . from i n p u t data, was  A vector,  obtained  added to the k ^ j m a t r i x to produce  the a c t u a l simultaneous equations r e q u i r e d to undergo multiple regression analysis. the f i n a l r e g r e s s i o n m a t r i x was illustrated  as  c ( q ) / c  c(£  n  )/c  The p(m  t o t a l dimensions of x n) and  can  be  follows:  o  K  K  o  K, In  n  K  2 1  K. 2n  3 1  K_  3n  K ml  K  mn Eq.  cU )/o 1  K  Q  c(K n )/c o  K  n  K, ln  v  K„  T h i s l a r g e m a t r i x was and  K  2 1  v  2n  K_  3 1  3n  K  K  ml  mn  then converted i n t o a s i n g l e v e c t o r  put on f i l e t o enable a c c e s s i n g by the  regression  multiple  routine.  ' Multiple The  regression  m u l t i p l e r e g r e s s i o n program o r i g i n a t e d from  Scientific  46  Subroutine Package developed by the  IBM  the  82.  Corporation  ( 3 ).  b e c a u s e i t was subroutine. as  I t was  s u i t a b l e f o r t h e MWD  specifically  The  designed  regression  incorporated  The  u s e d as  results included  d i d t h e Monroe p r o g r a m , and  t o remove i t .  t o be  a null  the  total  dictated  by  the  appearance of negative  a  intercept,  generated  routine  i n t o t h e p r o g r a m t o work on  of  an  s e t was  multiple regression  portions  calculation  was  each of the  regression matrix  p  (Equation  46),  frequencies  in  as the  solution.  P o s i t i v e 'constraint A  small  negative  the  was  used  to  scan f o r the  frequency r e s u l t i n g from the m u l t i p l e  calculation, to  subroutine  and  negative  t o l o c a t e t h e K.. -D value.  multiple regression through the  that  regression contributed  subroutine  controlled  calculation in its  progression  p parts of  This  vector  the  regression matrix.  to a l l those  shifting  the n e i g h b o r i n g  negative  vector  as  were e f f e c t i v e l y vector  follows:  i n t o the  the  When a  n e g a t i v e number was d i s c e r n e d , t h e a d d r e s s o f t h e K.. v e c t o r was n o t e d , and t h e v a l u e s i n t h e f i l e ID pertaining  largest  associated  removed  location of  by the  83.  c(  ? 1  )/c  K  o  n  K  " c(£  n  I f the K  )/c o  3 1  K  2 1  '  3 1  K  '  K, K_ ln 2n  K_ 3n  4 1  *  Eq. 47  K  4n  v e c t o r produced a negative  frequency, the  new matrix becomes: c(^)/c  o  K  n  K _ 2]  K  4 1  Eq. 4 8  c(£ s  The  n  )/c o  K. In  K ' 2n  K. 4n  dimensions o f the s e c t i o n o f t h e r e g r e s s i o n m a t r i x  to be analyzed  by the m u l t i p l e r e g r e s s i o n r o u t i n e were  then reduced a p p r o p r i a t e l y and the r e g r e s s i o n T h i s p r o c e s s was repeated u n t i l only p o s i t i v e were obtained.  repeated. frequencies  When t h i s was the case, the next r e g i o n •  of t h e r e g r e s s i o n m a t r i x underwent s i m i l a r treatment until  a l l p c y c l e s were completed.  frequencies  obtained  The r e s u l t i n g p o s i t i v e  from each c y c l e were used t o produce  the m o l e c u l a r weight d i s t r i b u t i o n . An example of t h e output o f the FORTRAN program w r i t t e n t o produce the complete MWD u s i n g r e g r e s s i o n can be seen i n F i g u r e p e r t a i n i n g to various  18.  multiple  Some comments  f e a t u r e s o f the output f o l l o w .  X-i-vi  : L v . ; , : ^ A . X j i . x a x J ! . . ' - X A j L . \ 7 _ o U - ^ y x,.i <,x.< x x » X ' t r X o o ; » A X X X A X A I O J U x a u U X U A A j J C ^ ^ ^  xxi • r ( n i c » » m n v i . i  . . " t f i-f.S  . . .  1 ..I  M . F . . . 3 . . T I L ! : . . . "  VI!.  J,.TlLf.  '  I H . . . « . . T l L t . .  OMVfcMSlTY  UF  H C  ,fr..ltl.f...7..TILh...0..l  CONFUTING  CE^IKE  1  ! I f : ' . . . 1 . . l i Ll-  T 3 ( A i< ] ? / )  0/! Jj:u<>  18.  Example  o f the output  multiple  T T T  1  T T T T T T T T T  T T '  T  T  I  T I  .  TT  7  MM M  nn ni'.  ou Oil  M ?1 '-1M  Off nr>  no fi'i  MM  M*  ?!M  :i  (Hi r)i«  no  OP n.»  T  T T  T  ,"i n  011 i' f 1 f 1' nH i if c ; t ; |  T TirrTr r T TTT  r  '!  r  M  MM  (.1  M  M;  M '••  M *•. MM  ?•! MM  MM  -1  (  S' M  in  H >;  IL.  '!  M •:  *  MM  I M T U I I I I 11  w  v v  T ! ! !  cc CC rr. cere rfrrrcrr  U t I  /  I I  / V  _ .  .  MM M«. M  .1  - .  -  >1  M  M>'  u 111 r iniinu?  STG «!lt<.H*S» ,  vv W  ...  W W  T  w  07t«9l.<U  StC.'ifP- CC  »t  LILL LL  '/ V  vv  i t  111  crrcecc  "Ie.<-<"  W  S  MMM  V!  .  r- y, M K MM ^ M \i>J  -Lt-1 T I M T-1-!  -cere  ,is£S>  Ms-  M  MM  I T  t.lAST  MM M f <M (,«!.# M  r "* (J un  nnr<(VH >iH OC-lO  c r cere r.rxccc cc :cr! cr cc cr. cc •:c  I'M  ,• M  M  I.LLLL'.U.l LLL I.LL'.t.LH LL1.L  rmj  07]  AP4  ?\ m  5 Nil"' - " 1  6  IT  .",:,T\  >,. -,i-c r CtCl.f? u  3  ..r*i.cuL«rION «HF«E:  THE  Pf"?T  CEI 1.  MCBE  •>:-.<C  tNA'.'T  C A L O ' L A T Jt'N  P.'.'J  ITT":'  P A ' Si'ECt'-MC vol.  j*rTK'>S  1 = TTTtWAT  't\t)f.  F0!»  T H J S  „T  C A L C ' I L A T  . an ii.n n-  ftlN  I ' I M  f-ODt.  T'-.J » » *  c  Figure  ? = SI<OOTHNI'IG . MCOF.  A')F ! -liFNTSFt'S 1 ,(Jrt(l(11  > / > A \ f i t n': 11.' . . . 1 . . ' J l  .  .  .  I i  ? > / / !  o f the  regression T4KD -  s  program.  1  ,•„•  i >.'.,.  V .  85.  n  86.  - <r o c *•"»  if  O -ftj4 > C CWOK C " " O o — C — O C r * > ff C I T S  7) 3 O C I 1 * O A) iirtj : £ j ^ O O A.  « *  * *  Ui U. bit  ^ Q tr u. c 3 u; W o « u. 3 •Ji — '.2 > ui v j • •J: i o u. a. a  a  »_ rz< < ;• r-  u.  rv o ru o  rr — o o c c: s c c = e c  — -* = = c C C 3  o * -O O IT - 3 O  o  3  C f\J — U " > — a. o c = 1 -c —  -1  c ~^ f  LOI 3  Z  o  UJ  <>  ^  O -t «* U- c »- -» OUJ < > •AJ M ia tt a > c O u . MJ <4 * a ^ ; u o u.  4  C 1 3  •*  »- UJ lfl UJ n o cn i  ul -A O Z  I c tt «  . u. z  »- ' o « * Z UJ 5' O *a »-» •» UL O »- —  c w •> > U J  U J  a a > c a o u. ui UJ a c c aiuoii u; o * •-co -* 7, r<- ut tr.  UJ  THE  SFRIES I S -  TMF  Hat'.  FREQUENCIES  -S-.P. OF FBCnilEnC TFS Slix  IS«  O F FREQUENCIES f S -  0,27660  0.Ont, 8.7  0.01 U13  B.lltM  n^m^u  I S - ..  f)F S T A T I S T I C A L  =- gr-"^i  r  T  . 0.00102. 5'. 0 0 0 0 ll 0 . 00002  BFCM  -()'. onaau-  THE FREQUENCIES S.n,  ARE* i s -  OF FRFMIENCIES  . SUM. OF  .  FRF.QUEN.CIES  IS-...".  STD  FRP08  DEP  OF  n  .  nr nFvjfATTOM  HE AN  T  H L I L T CORK COEF S U M S U OUE TO REGN_.MFAN  S O DF  2.8027? 1 o. no (mo ?To'i u i . o n  . DEG UK FREEDOM FwVAl tIK  OF.... CYCLE  NUMBER  2 .  0,0761 I  0,•5(1616  0.32305  o ,<ioi sa  o.a?3?o  O.0O6S0  0.00718  0.00587  0,00271  o.oo?06 1.900711  1.00000 11.0 1362 .  S S  .. 3 7 6 8 1 .  . 6<I058.  108808.  .  Of  STATISTIC*!  3.00.016. O.oon/iO 5'. 0 0 o f, o ll'. il 0 0 (10 ..  :  panu  ---  _ 22165,.  nF E S T  FRFFOOM .  PARAMETERS :  . 13038..  • TABLE  . INTER.CE.PT  »,»»»»  1.89285  END  M.ri S F K I E S  0,63006  0.S2U88  IMTERCFPT  THE  "Sis'*.  0.U2236  TA«LF  Mg»N SM  56000.  0.0J8<>5  »=a -ST(1 EHROI) O F E S T DEC. OF FI)E t *0M ss O F o E v i A T i r i M rpo« PE-V1AT lUtlS  33000.  prr..  o.ooooo  S O OE V I ATIONS  ENfi  OF  PAUAMFTFRS  .H'JLT  CORK  ... l.O"o?'j li.ot 2.fi027S 19.PP nop .1 1 9l)5 J3<l .54  .COEF  SU* Su. DUE TO REGN MEAN SU OF SS 0 F « OF FHFFOOM F-VALUE  CYCLE  NUrtBEH  3  lllllllllll.ll.l.........».>.»l.>l.ll .|. || | l|<).,|Hlll('l ltl)lt>>U»t>l>»l|»( l|l»»ll«l<ll>»IXl|HIIHIHII<l <  THE  Hn S F P I F 3 I S -  TrtF  F9"f.flnFpiiC IFS A HE.  S.n, SUM  OF FREOHENCIFS I S . OF FRFOMEMCIES I S -  I AflBB. 0 . 1 7533 0,00305  1  t  )  25310. n.hPStlT . o,oo6sn  <  (13026.  <  731(15.  0.2161 I  n . t i o i 31  n.oonii  0,00679  I.90/108  TAPLF  |iF S T A T I S T I C A L  PARAMETERS  I21306, P.2567lV. 0,00310  88..  I • •  i c. -c r- c C Klrt C 3 ' c *- c e x c e a c- c  ij  i  <  2 ! ! U 2  U. £} »- e u£ •« > »- a a :> c • a C' u u_ i w a | c a  :  j  cr  u: O u.  > cd < l 7 f- U,' « w i  «n c  x  89.  1. define  A t a b l e of p e r t i n e n t parameters was p r i n t e d t o the v a r i a b l e s used i n the c a l c u l a t i o n .  2. and  The raw data used f o r the c a l c u l a t i o n , X ,£  c(^)/c  o  purposes.  were p r i n t e d f o r r e f e r e n c e  and checking  A f o u r t h c o n s t a n t was a l s o p r i n t e d t o i d e n t i f y  whether the data were from s c h l i e r e n o r UV o p t i c s , so as to p r o c e s s the data a c c o r d i n g l y . were i n the same format r e q u i r e d programming program. end  The data as p r e s e n t e d f o r Scholte's  linear  A zero s e t was always put i n a t the  o f the data s e t t o a l l o w the g e n e r a t i o n o f the dummy  v a r i a b l e s which e f f e c t i v e l y removed the i n t e r c e p t . 3.  F o r convenience, the i n d i v i d u a l molecular weight  values pertinent 4.  to each r e g r e s s i o n  The f r e q u e n c i e s ,  pertaining  were p r i n t e d .  o r the r e g r e s s i o n c o e f f i c i e n t s ,  t o the corresponding molecular weights i n the  s e r i e s were p r i n t e d . 5.  The sum of the r e g r e s s i o n  p r i n t e d , because i t was a v a l u a b l e chosen s e r i e s was 6.  c o e f f i c i e n t s was i n d i c a t o r o f whether t h e  correct.  A table of s t a t i s t i c a l  parameters allowed a  judgment of the f i t a t t r i b u t e d t o the s e r i e s used i n the regression. 7. the  I f a negative frequency appeared, the l o c a t i o n o f  l a r g e s t one was p r i n t e d . This  l o c a t i o n was always i n  r e l a t i o n t o the number o f m o l e c u l a r weights used i n the  90.  ^egression.  T h e r e f o r e , the message "Molecular weight  number 4 i s n e g a t i v e and w i l l be removed" r e f e r r e d t o the f o u r t h m o l e c u l a r weight i n t h e o r i g i n a l s e r i e s . next r e g r e s s i o n s t i l l produced  a n e g a t i v e frequency, i t s  l o c a t i o n would then be p e r t i n e n t t o the reduced used i n the l a s t r e g r e s s i o n .  I f the  T h i s can be b e s t  series illustrated  as f o l l o w s : Assuming 7 m o l e c u l a r weights  were used  f o r the  regression:  f  l l M  f  2 2 M  f  3 3 M  f  4 4  ~ 5 5  M  f  M  f  M 6  6  f  7 7 M  E q  '  4  9  f^Mj. was n e g a t i v e f o r the f i r s t r e g r e s s i o n and t h e r e f o r e removed.  f  l l M  The next s e r i e s t o be r e g r e s s e d would be:  f  2 2 M  f  3 3 M  f  4 4 M  f  6 6 M  f  7 7 M  E  q  '  5  0  T h e r e f o r e , i f a n e g a t i v e frequency o c c u r r e d i n t h i s r e g r e s s i o n f o r f^-M-, i t would be l a b e l l e d as the removal D  D  o f m o l e c u l a r weight 5.  T h i s f a c t s h o u l d be kept i n mind  when a s s o c i a t i n g the p o s i t i v e f r e q u e n c i e s w i t h the o r i g i n a l m o l e c u l a r weight 8.  series.  Whenever the c a l c u l a t i o n had reached a p o s i t i v e  s o l u t i o n , the c y c l e number p was p r i n t e d , a l o n g w i t h two l i n e s o f s t a r s t o separate the c o n c l u s i o n s o f the i n d i v i d u a l cycles.  91.  Comparison of L i n e a r Programming and M u l t i p l e R e g r e s s i o n  Once the FORTRAN v e r s i o n of the m u l t i p l e r e g r e s s i o n program was approaches  completed, was  a d i r e c t comparison  readily feasible.  a wide v a r i e t y of models  (2 and  be concluded t h a t the two identical results.  3 component) i t c o u l d  c a l c u l a t i o n s produced  s e r i e s , the i n t e r v a l and the  t h a t produced  range  In o r d e r t o  i l l u s t r a t e the e f f e c t of the n e g a t i v e molecular and how  almost  comparisons,  of each c a l c u l a t i o n had t o be i d e n t i c a l .  removal  two  From the a n a l y s i s of  To make these d i r e c t  the m o l e c u l a r weight  o f the  weight  i t l e d t o almost the same s o l u t i o n as  by l i n e a r programming, an example i s  presented i n Table I I I .  The f(M)  column i n T a b l e I I I  r e f e r s t o the second r e g r e s s i o n of the K..  matrix  that  13  was  r e q u i r e d any time a n e g a t i v e frequency  Since any molecular weight frequency was  appeared.  associated with a negative  removed from the r e g r e s s i o n , i t c o u l d be  a s s i g n e d a zero frequency v a l u e .  In comparing the  frequency v a l u e s obtained by l i n e a r programming and  by  m u l t i p l e r e g r e s s i o n , a very c l o s e agreement between the two  s o l u t i o n s was  observed.  The a n a l y s i s of more  complex systems showed s i m i l a r  results.  92.  Table  III.  Comparison o f F r e q u e n c i e s R e s u l t i n g from t h e A n a l y s i s o f a Two C o m p o n e n t S y s t e m * U s i n g M u l t i p l e R e g r e s s i o n and L i n e a r Programming  Multiple  Molecular Weight (daltons)  f (M).  Regression  1  f(M)  2  Linear Programming f (M)  11,000  0.17  —  0.18  22,000  0.02  -  0.00  44,000  0.46  0.48  88,000  0.33  -  13,081  0.33  0.21  0.21  26,163  -0.20  0. 00  0.00  52,325  0.71  0.61  0.61  104,650  0.15  0.16  0.16  15,556  0.33  0.29  0.29  31,113  -0.07  0.00  0.00  62,225  0.67  0.63 .  0.63  124,415  0.06  0.06  0.06  18,500  0.32  —  0.32  36,999  0.12  -  0.11  73,999  0.53  -  0.54  147,998  0.01  —  0.01  *20,000 - 80,000 speed data  dalton mixture  (1:1 r a t i o ) ;  0.33  single  F a c t o r s A f f e c t i n g the M o l e c u l a r Weight  Distribution  Once i t became apparent t h a t the m u l t i p l e  regression  program c o u l d produce s i m i l a r r e s u l t s to l i n e a r programming, a study was undertaken t o i n v e s t i g a t e the e f f e c t s o f speed, m o l e c u l a r weight i n t e r v a l and m o l e c u l a r weight range on the  d i s t r i b u t i o n obtained.  In h i s work, S c h o l t e concluded  t h a t m u l t i p l e speeds and a m o l e c u l a r weight i n t e r v a l o f 2.0 were the most s u i t a b l e c r i t e r i a systems.  f o r the a n a l y s i s o f p o l y m e r i c  However, p r o t e i n systems are comprised o f  d i s c r e t e m o l e c u l a r weight components r a t h e r than continuous distributions. resentative  The simple Rinde expansion model i s r e p -  o f a mixture o f d i s c r e t e components, but  a n a l y s i s o f data generated from t h i s model produced a continuous d i s t r i b u t i o n . of  T h e r e f o r e , even an i d e a l mixture  homogeneous p r o t e i n s , not s u b j e c t t o e x p e r i m e n t a l e r r o r  would produce a d i s t r i b u t i o n , because the c a l c u l a t i o n not  r e s o l v e an i n f i n i t e l y narrow d i s t r i b u t i o n .  could  Upon  a n a l y s i s o f l o g normal model d a t a , w i t h v a r i o u s s t a n d a r d d e v i a t i o n s , i t was  found t h a t the d i s t r i b u t i o n o b t a i n e d  from a standard d e v i a t i o n of 0.05  was  essentially i n -  d i s c e r n a b l e from t h a t of the Rinde model.  In g e n e r a l ,  broader d i s t r i b u t i o n s were e a s i e r t o reproduce than narrower ones which were more demanding i n the c o r r e c t c h o i c e o f speed, i n t e r v a l and range.  S i n c e the Rinde expansion i s  a l i m i t i n g case, and r e l a t i v e l y easy to use,  i t was  i n i t i a l l y used to screen the e f f e c t s o f the v a r i a b l e s on the d i s t r i b u t i o n . apparent  In the ensuing  study i t became  t h a t the e f f e c t s of the v a r i a b l e s were not  independent of each other, and i n t e r a c t e d i n a complex fashion.  A.  ' Speed  Speed was  a critical  f a c t o r i n the p r o d u c t i o n of a  w e l l d e f i n e d molecular weight d i s t r i b u t i o n .  By a p p l y i n g  i n c r e a s i n g speeds, f r a c t i o n a t i o n of the sample o c c u r r e d , a l l o w i n g the v a r i o u s components to c o n t r i b u t e t o the d i s t r i b u t i o n . At low speeds the h i g h e r m o l e c u l a r  weights  were the major c o n t r i b u t o r s to the c o n c e n t r a t i o n d i s t r i b u t i o n i n the c e l l .  At high speeds these would be  r e l e g a t e d to the bottom of the c e l l  a l l o w i n g the  m o l e c u l a r weight components to c o n t r i b u t e . were encountered  low  No problems-  i n reproducing the d i s t r i b u t i o n o f  two  and t h r e e component systems when a wide range of speeds (six to  speeds ranging from 10,000 to 30,000 rpm)  were used  e v a l u a t e model data, u s i n g f o u r data p o i n t s per speed  as advocated  by S c h o l t e .  However, s i n c e more data p o i n t s  were r e a d i l y a v a i l a b l e from the UV  scanner,  a reduction  i n the number of speeds, with a c o n c u r r e n t i n c r e a s e i n data  p o i n t s was  attempted.  In g e n e r a l , i t was  found t h a t the  number of speeds c o u l d be reduced t o one speed per component, if  20 data p o i n t s were used and the speeds were chosen  judiciously.  I f the number of speeds chosen  i n s u f f i c i e n t or d i d not a l l o w a s u f f i c i e n t  was  contribution  from the i n d i v i d u a l components, an 'undefined' d i s t r i b u t i o n was  produced. Examples  o f a d e f i n e d and u n d e f i n e d d i s t r i b u t i o n f o r  a two component system can be seen i n F i g u r e s 15 and 19 respectively.  The undefined d i s t r i b u t i o n had  several  f e a t u r e s which made i t r e a d i l y r e c o g n i z a b l e . These were a f l a t topped and n o n - b e l l shaped peak f o r the lower molecular weight component and a s h i f t of the h i g h m o l e c u l a r weight peak to a lower m o l e c u l a r weight than o r i g i n a l l y d e f i n e d by the model d i s t r i b u t i o n .  The term  'undefined' a l s o c a r r i e d a d d i t i o n a l s i g n i f i c a n c e i n t h a t the t r u e r a t i o s of the components were l o s t , and the low m o l e c u l a r weight component v a r i e d i n i t s l o c a t i o n on the x - a x i s , i f the range was  changed.  A similar  situation  e x i s t e d i n the case of t h r e e component systems.  In a l l  c a s e s , even i f only one speed was used, the r e s u l t i n g undefined d i s t r i b u t i o n i l l u s t r a t e d the number of components p r e s e n t i n the system, i f they were s u f f i c i e n t l y separated i n m o l e c u l a r weight  (Figure 20).  speed d i s t r i b u t i o n s was  T h i s p r o p e r t y of s i n g l e  l a t e r used advantageously.  96.  0.8  0.6 ©  S  0.4 ©  0.2  r  0  ©  ©  0.0 14  t ,.Q. 28  56  112  224  M x 10" F i g u r e 19.  S e n r i l o g a r i t h m i c p l o t o f f (M) v s . M f o r a 25,00080,000 d a l t o n m i x t u r e (1:1 r a t i o ) . S i n g l e speed data; i n t e r v a l =2.0.  0.6  0.4  I-  5 0.2  0.0 M x 10 F i g u r e 20.  Sendlc>garit±imic p l o t o f f (M) v s . M f o r a 25,00080,000-320,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . S i n g l e speed d a t a ; i n t e r v a l = 2 . 0 .  B.  Interval  The  effect  interval the  and  o f c h o o s i n g an a p p r o p r i a t e m o l e c u l a r w e i g h t  and r a n g e c o u l d  b e s t p o s s i b l e MWD.  t h a t an i n t e r v a l optimum. study, the  since  true  t o be u s e d was  dependent  series.  was  upon  That i s , i f the range,  u s u a l l y b e s t d e s c r i b e d by u s i n g  The  interval  An  c o u l d be c h a n g e d  comparing  Figures  shape  15  and  21.  interval  Figure  c a n be  21 u s e d  self-association  22 t o 24 r e p r e s e n t  o f b o v i n e serum  equilibrium  i n a r a t i o o f 2:1.  illustrated  how  of  t h e two  of  1.4  components *  resolved  Adjustment  an i n t e r v a l  two  o f 2.0  first  resolution  (BSA)  at  distribution  u s i n g an  b u t t h e y were  reduced  a model o f  a l l o w e d no  The n e x t one,  peaks,  of the range  The  albumin  seen  t h e same  c o u l d be o b s e r v e d when t h e i n t e r v a l was Figures  suit  was  15 and an o b v i o u s improvement i n  t o 1.5.  a  by  example o f t h e improvement w h i c h  by c h o o s i n g t h e a p p r o p r i a t e  was  in this  t h e c o n s t a n t o r t h e power v a l u e o r b o t h t o  system.  f r o m 2.0 the  f o u n d t o be  study covered only a r e l a t i v e l y short  d a t a as F i g u r e and  between t h e m o l e c u l a r w e i g h t s  the i n t e r v a l  interval.  possible by  o f 2.0  distribution  altering  i n obtaining  S c h o l t e c o n c l u d e d f r o m h i s work  range of the m o l e c u l a r weight  smaller  the  be v e r y b e n e f i c i a l  H i s a s s u m p t i o n was  system under the  range  resolution interval  distorted.  t o s u i t t h e s y s t e m more  closely  98.  1.0  0.6  V  0.2  M x 10 Figure 21. Semilogarithmic plot o f f CM) vs. M for a 25,00080,000 dalton mixture (1:1 ratio). Multispeed data; interval = 1.5.  0.8  ' ~~ G  0.6  0.4 © 0.2  \  0  © 0  0.0 30  Figure 22.  60  120  240  -3 M x 10  Semilogarithmic plot of f CM) v s . M for a 67,000134,000 dalton mixture (2:1 ratio). Multispeed data; interval = 2.0.  30  42  .  58  82  115  M x 10~ F i g u r e 23.  161  225  3  S e m l c g a r i t h r t d c p l o t o f f (M) v s . M f o r a 67,000134,000 d a l t o n m i x t u r e (2:1 r a t i o ) . M u l t i s p e e d d a t a ; i n t e r v a l - 1.4.  0.8 ©• ©  0  \ >  /  \  /  \  /  ©  ©  /  *  ©  \ • •  30  42  58  82  Q  -3  /  0  * •  M x 10 F i g u r e 24.  •  N  .  \  ©  / Q  I  115  I  161  2.  225  S e r r d l o g a r i t t a i c p l o t o f f (M) v s . M f o r a 67,000134,000 d a l t o n m i x t u r e (2:1 r a t i o ) . M u l t i s p e e d d a t a ; i n t e r v a l = 1.4, w i t h t h e r a n g e h a v i n g been adjusted.  100.  produced a t h i r d d i s t r i b u t i o n which had the d e s i r e d c h a r a c t e r i s t i c s of a good MWD,  r e f l e c t i n g the c o r r e c t  p r o p o r t i o n s of the components along w i t h the c o r r e c t weight average molecular weight.  C.  Loss of data  One  other f a c t o r i n v e s t i g a t e d was  removing h i g h c(£)/c T h i s had  the e f f e c t of  v a l u e s from the s y n t h e t i c data.  to be c o n s i d e r e d because a t higher speeds the  model C ( C ) / C  q  v a l u e s r e a d i l y a t t a i n e d much l a r g e r  than c o u l d be r e a l i s t i c a l l y e v a l u a t e d by any system.  values  optical  Removal o f these v a l u e s u s u a l l y d i d not  affect  the d i s t r i b u t i o n , as long as t h e r e were enough o t h e r data points available.  The only d i s t r i b u t i o n s a f f e c t e d  removal of c ( / c  v a l u e s were those i n v o l v i n g  by  single  speed data d e r i v e d from too high a speed.  D.  Summary•  The  d i s t r i b u t i o n s d i s c u s s e d were a minor p a r t of  the t o t a l number o f v a r i a t i o n s and combinations i n order to understand distribution. the r e s u l t s was  how  studied  the v a r i a b l e s a f f e c t e d  the  The major c o n c l u s i o n to be d e r i v e d from t h a t some p r i o r knowledge of the  molecular  101.  w e i g h t s o f t h e components w o u l d the  correct  Thus, had  combination of speed,  i n assessing  t o be  A.  Up  statistical  insight  point,  parameters i n t h e MWD  available  the major  (30).  the  f i t o f any m o l e c u l a r w e i g h t  the  complete  parameters would  t u r n e d o u t t o be  chosen  He  of  f o r pursuing the  had  suggested  that  allow the e v a l u a t i o n series true.  to a  of  distribution.  I f the molecular system d i d not cover  completely out of l i n e with the  t h e system,  noticeable reduction  multiple  The a v a i l a b i l i t y  for a particular  r a n g e , o r was  w e i g h t s m a k i n g up  the most  through the use o f  rationale  statistical  series  into  has b e e n made o f t h e  calculation.  the  weight  range.  parameters  no m e n t i o n  c o n c e p t p u t f o r w a r d by Magar  this  and  R e g r e s s i o n Approach  the s t a t i s t i c a l '  t h e s e p a r a m e t e r s was  Basically,  interval  interval.  the M u l t i p l e  to t h i s  regression  to gain  r a n g e and  'Assessing  i n determining  e x p e r i m e n t a l d a t a , a number o f MWDs  calculated  appropriate  Expanding  be h e l p f u l  a much p o o r e r and v e r y  i n f i t was  noted.  However, when t h e  m o l e c u l a r weights covered the complete range,  for a l l  practical  discerned  purposes  little  difference  between a s e r i e s w h i c h b e t t e r  suited  c o u l d be  t h e s y s t e m and  a  102.  series also  w h i c h was  found  t o be  poorer.  The  sum  a good i n d i c a t o r  of the  frequencies  was  o f whether the range  was  w i d e enough, w i t h a v a r i a b l e v a l u e b e i n g o b t a i n e d when t h e r a n g e was  insufficient  when t h e r a n g e was distribution, statistical correct  B.  parameters being  concept  Although in  The  o f an  the  'iterative  statistical  However, d u r i n g t h e  values  i t was  present. weight  noted  a mixture 20,000 and then  the  p a r a m e t e r s were range f o r the  helpful  calculation,  somewhat d i s a p p o i n t i n g .  of the  appeared  individual  be  illustrated  molecular  the o t h e r  distributions  the  frequency  were  directly  components  o n l y when t h e  coincidental with  molecular  t h e maxima o f This  i n the  t h a t m o d e l d a t a had  o f two  to assess  mainly  solution  d i s t r i b u t i o n under i n v e s t i g a t i o n .  Assuming  the  the molecular weights  This situation  association w i l l  plot, with  t h a t i n some c a s e s  to the r a t i o s  s e r i e s was  o f a good  a n a l y s i s o f t h e many m o d e l  associated with  proportional  obtained  range.  a c t u a l u s e f u l n e s s was  calculated,  major c r i t e r i o n  used  the c h o i c e o f the c o r r e c t  their  a constant being  t h e r e f o r e , became t h e MWD  choice of  The  good.  and  the  coincidental f o l l o w i n g example.  been generated  for  w e i g h t components, one  80,000 d a l t o n s , i n a r a t i o  a coincidental molecular  weight  of  s e r i e s would  of 1:1, be:  103.  104.  10,000  20,000  40,000  80,000  and the c o i n c i d e n t m o l e c u l a r weights would be 20,000 and 80,000 d a l t o n s . The for  frequency v a l u e s o b t a i n e d from the MWD  these molecular weights  components.  calculation  would be 0.50 f o r both  T h i s was the case whether m u l t i s p e e d o r s i n g l e  speed data were used.  This observation implied that  unispeed d a t a , i f s o l v e d o n l y f o r the two maxima r a t h e r than a s e r i e s , would a l l o w the q u a n t i t a t i o n o f the r a t i o of  the two components.  In the case o f m u l t i s p e e d  data,  a reasonable f a c s i m i l e o f the t r u e d i s t r i b u t i o n c o u l d be o b t a i n e d , t h a t r e l a t e d t o the r a t i o o f the components d i r e c t l y by the r e l a t i v e area under the peaks. n o t p o s s i b l e w i t h unispeed  s i n c e an undefined  T h i s was distribution  was g e n e r a l l y produced. By u s i n g the MWD  program t o s o l v e f o r 20,000 and 80,000  d a l t o n s only, and t e r m i n a t i n g the c a l c u l a t i o n a f t e r one c y c l e , the two frequency v a l u e s o b t a i n e d were s t i l l 0.50. fit.  The s t a t i s t i c a l  both  parameters i n d i c a t e d an e x c e l l e n t  Any other combination  o f m o l e c u l a r weights  produced  a s i g n i f i c a n t l y poorer f i t , as judged by t h e s t a t i s t i c a l parameters  (F-ratio).  Some r e s u l t s o f t h i s  can be seen i n Table IV.  calculation  I n v e s t i g a t i o n o f t h r e e component  systems i n d i c a t e d t h a t s i m i l a r q u a n t i t a t i o n from  single  speed data was p o s s i b l e .  c  105.  Table  IV.  F r e q u e n c i e s a n d F - R a t i o s O b t a i n e d By M u l t i p l e R e g r e s s i o n A n a l y s i s o f a 20,000-80,000 D a l t o n Mixture (1:1 R a t i o ) U s i n g S e l e c t e d Molecular Weight P a i r s ,  M o l e c u l a r Weight Pair  Frequencies  82,000  0.46  22,000  0.53  80,000*  0.50  20,000  0.50  85,000  0.42  25,000  0.56  85,000  0.48  16,000  0.53  71,000  0.63  16,000  0.37  65,000  0.67  20,000  0.33  *  Best  f i t solution  F-Ratio  2.0  x  10  8.6  x  10  2.0  x  10  7.7  x  10  1.7  x  10  2.8  x  10  6  7  5  3  4  3  106.  Two  and  quantified into  three  by  component m o d e l s y s t e m s c o u l d , be  entering  the  the r e g r e s s i o n equation.  observation  t h a t the  values,  l e d to the  a guide  describe iterative single find  to f i n d i n g  a system.  corresponding  C.  The  the  the  b a s e d on  initial  iterative  the  f e a t u r e was  component p r e s e n t . a simple  be  described  parameters  the  weights  decided  that  characteristics c o u l d be  weights,  and  an  of a .  used  hence  to  to  their  algorithm  shift on  had  characteristic  the m o l e c u l a r  weight  obtained  algorithm  w e i g h t s t o be a guide.  i n p o i n t form,  axis.  as  that allowed  searched The  molecular  a  basis  the  for, using  basic algorithm  for a three  The  f o r each  T h e s e f a c t s were u t i l i z e d  F - r a t i o as  features,  of the weight average  t h a t a peak was  incremental  correct molecular statistical  i t was  weight  ratios.  weight to lower values  for  these  distribution  f i t molecular  o f w h i c h was  other  the  parameters  correct molecular  Undefined d i s t r i b u t i o n s one  and  correct molecular  Initially,  algorithm  best  fact,  concept of using  speed u n d e f i n e d  the  This  weights  regression s t a t i s t i c a l  were good i n d i c a t o r s o f  as  c o r r e c t molecular  component  the will  system.  107.  1.  The w e i g h t a v e r a g e m o l e c u l a r  calculated  from the o r i g i n a l  distribution using  obtained  was  molecular  f r o m a c o m p l e t e MWD  weight  calculation,  the following r e l a t i o n :  y M  M  =  . f. —  E f 2.  E q . 51  ±  The w e i g h t a v e r a g e m o l e c u l a r  run  through  the  resulting F-ratio  weight values  one c y c l e o f t h e r e g r e s s i o n c a l c u l a t i o n  (M^) component was and  undefined  weight  regressed  stored.  then  again.  The h i g h m o l e c u l a r  incremented  The F - r a t i o  from  i f g r e a t e r , t h e M^ v a l u e was  procedure  continued  maximization,  t h e M2 was  maximized again, going  until  through  then  incremented  c o n t r o l was  a l l three molecular  was h a l v e d  expected, after twenty  again.  passed  t o M^.  weights,  the F - r a t i o After  control  w e i g h t component, t h e  passed increment  As :;.  w e i g h t c h a n g e s became v e r y  a number o f c y c l e s .  In g e n e r a l ,  This  After  by CLM2 u n t i l  and t h e a l g o r i t h m was r e p e a t e d .  the molecular  value,  the F - r a t i o maximized.  back t o the h i g h molecular dM^  incremented  dM^  this  r e g r e s s i o n was compared t o t h a t o f t h e p r e v i o u s and,  and  weight  by some v a l u e  obtained  were  small  approximately  i t e r a t i o n s were r e q u i r e d t o a l l o w c o n v e r g e n c e t o  the b e s t  f i t molecular  originally  weights.  This algorithm  t e s t e d on t h e Monroe 1880 and p r o v e d  sufficiently  promising  t o be programmed  was t o be  i n t h e FORTRAN  108.  version  for further testing  on  the  IBM  a s s o r t m e n t o f m o d e l s was  t e s t e d and  f o u n d t o work r e a s o n a b l y  well  algorithm was  c o u l d be  fact  The  the  of the  initial  that a search  simplex  previous true  first  from which  iterative  method  s e c t i o n allowed  t h a t the  values, search  procedure  i t had was  two  major  w e i g h t i n c r e m e n t had  l a r g e an  i t .  a m u l t i v a r i a t e a p p r o a c h w o u l d be solving  f o r the  In r e c e n t  frequencies years,  Box  and  procedure  (5), who  (EVOP) o f  a combination  of  searching  factorial  f o r the  design  the  and  care  optimum  decided  that  of  optimization  initially the  the  Furthermore,  have n o t been  developed  and  MWDs.  multifactor iterative  M u l t i f a c t o r a n a l y s i s was Draper  limitations..  a b e t t e r method  p r o c e d u r e s have been developed, b u t frequently.  I t was  of undefined  estimates  chosen with  increment could overshoot  small might never reach  described  unidirectional,  t o be  too  frequency  c o n v e r g e n c e t o good  the molecular s i n c e too  unde-  algorithm  univariate iteration  frequency  was  semi-quantitative  independent.  by  was  This  s e c o n d t h a t t h e v a r i a b l e s were n o t  and  varied  obtained.  A l t h o u g h the  The  algorithm  i n most c a s e s .  to e s t a b l i s h the  distributions,  values  in  the  A  a f e a s i b l e a p p r o a c h t o s o l v i n g f o r t h e maxima o f  fined  D.  served  computer.  used  proposed  evolutionary  optimum by regression  operations  using techniques.  109.  A major advance over the EVOP was the simplex o p t i m i z a t i o n method developed by Spendlay e t a l . (51).  T h i s was an  e m p i r i c a l i t e r a t i v e feedback c a l c u l a t i o n t h a t was intuitively  simple i n concept and o p e r a t i o n .  o r i g i n a l method was improved  by Nelder and Mead  f u r t h e r m o d i f i e d by Morgan and Deming o p t i m i z a t i o n had a unique advantage approaches  because  This (37) and  (33). The simplex  over the s i n g l e  factor  of i t s a b i l i t y t o overcome the r i d g e  phenomenon t h a t c o u l d prevent o p t i m i z a t i o n due t o v a r i a b l e interaction.  Furthermore,  the simplex approach  allowed  the d e f i n i t i o n of a range over which a m u l t i d i r e c t i o n a l search f o r the optimum c o u l d be i n s t i g a t e d , w i t h a l l the v a r i a b l e s changing s i m u l t a n e o u s l y .  1.  The simplex method  The simplex procedure has been d e s c r i b e d i n s e v e r a l p u b l i c a t i o n s , w i t h the most e a s i l y grasped presented by Morgan and Deming d e s c r i p t i o n lacked c l a r i t y example c a l c u l a t i o n .  version  (33). However, even t h e i r  as i t d i d not show a simple  In the f o l l o w i n g pages,  the concept,  o p e r a t i o n and c a l c u l a t i o n o f the simplex w i l l be p r e s e n t e d i n r e l a t i o n t o i t s use i n s e a r c h i n g f o r the optimum o f an undefined MWD. demonstrated,  Once the procedure has been f i r m l y a f l o w c h a r t w i l l b e s t i l l u s t r a t e the  r e p e t i t i v e nature o f the r e s t o f the a l g o r i t h m .  110.  The simplex concept can be r e p r e s e n t e d by a geometric f i g u r e composed o f one more dimension  than the o r i g i n a l  number o f f a c t o r s under i n v e s t i g a t i o n .  Thus, f o r a two  v a r i a b l e system the simplex i s r e p r e s e n t e d by a t r i a n g l e , for  a t h r e e v a r i a b l e system,  a t e t r a h e d r o n , and so f o r t h .  The v e r t i c e s o f these geometric response  (dependent  v a r i a b l e ) t o the c o n t r i b u t i n g  For i l l u s t r a t i v e be d i s c u s s e d . F  b  F  n w F  r  e  P  r  e  s  f i g u r e s i n d i c a t e the  purposes  factors.  a two f a c t o r system w i l l  . .  R e f e r r i n g t o F i g u r e 25, the t r i a n g l e e  n  t  s  t  n  e  s t a r t i n g simplex f o r an o p t i m i z a t i o n  search o f a two component system. F^ r e p r e s e n t the F - r a t i o s  The v e r t i c e s F.  F , and  (response) o b t a i n e d by r e g r e s s i n g  the t h r e e s e t s o f two molecular weights  ( f a c t o r s ) . These  F - r a t i o s are the b a s i s f o r d e t e r m i n i n g s u c c e s s i v e simplexes, thereby p r o p a g a t i n g a search f o r the b e s t f i t combination of m o l e c u l a r weights.  I g n o r i n g a t t h i s time how t h e  o r i g i n a l f a c t o r s were chosen,  the m o l e c u l a r weights were  put i n t o m a t r i x form as f o l l o w s :  10,000  60,000  29,319  67,765  15,17 6  88,97 8  r e g r e s s i o n. :  F  w Eq. 52  F  n  0  L 50  60  70  80 M x  F i g u r e 25.  10  -3  90  I l l u s t r a t i o n o f the f i r s t basic operations routine.  100  o f the simplex  110  120  optimization  112.  The  simplex  algorithm required  representing  the v e r t i c e s  undergo s p e c i f i c and  that  this  of the s t a r t i n g  geometric  c  Fn referred  (F-ratio). '  to the worst,  respectively.  was t h e c a l c u l a t i o n responsive  where  matrix, would expansion  on t h e e v a l u a t i o n  The F - r a t i o v a l u e s F , F, a n d w b b e s t and n e x t  The i n i t i a l  simplex,  changes o f r e f l e c t i o n ,  v a r i o u s forms o f c o n t r a c t i o n based  of the response  starting  to best values  step i n the simplex a l g o r i t h m  of the centroid,  t h e mean o f t h e more  factors:  C =  (B + "N)/2  C -  centroid  E q . 53  B - best molecular  weight  N - mean o f a l l m o l e c u l a r w e i g h t s other  than  i n the vector,  t h e b e s t and w o r s t .  therefore:  C C  The  ±  2  =  (67,765 + 8 8 , 9 7 8 ) / 2 = 78,371  = (29,319 + 15,176)/2 = 22,247  c e n t r o i d w o u l d t h e n be l o c a t e d  centroid mirror  v a l u e s were u s e d  image o f t h e w o r s t  following  a t C^, C^.  to generate location,  These  a reflection or  according to the  relation:  R = C +  (C - W)  Eq.54  113.  where W r e p r e s e n t s  the worst molecular  weight o f each s e t .  = 78,371 +  (78,371 - 60,000) =  96,742  = 22,247 +  (.22,247 - 10,000) =  34,494  Therefore:  R  ±  R  The  2  reflected  represented regressed  values  occupied  a new m o l e c u l a r  t o o b t a i n a new  was g r e a t e r t h a n  the p o s i t i o n  R^,  R  2  and  w e i g h t s e t t h a t c o u l d be  response  F^, an e x p a n s i o n  F .  I f the F  was p e r f o r m e d  value according  to the f o l l o w i n g r e l a t i o n :  E = C + 2 (C -  W)  Eq.  55  Therefore,  The  E  1  = 78,371 + 2(78,371 - 60,000) = 115,113  E  2  = 22,247 + 2(22,247 - 10,000) =  e x p a n d e d v a l u e s became E-^, E  these molecular than  F  r  response  2  and t h e r e g r e s s i o n o f  weights produced F .  then molecular molecular  w e i g h t s F^,  If F  115,113  29,319  67,765  15,17 6  88,978  g  was  greater  replaced the worst  weights i n the matrix  46,741  46,741  as f o l l o w s :  Eq.  56  114.  T h i s m a t r i x was used t o c a l c u l a t e a new c e n t r o i d and the e n t i r e process repeated.  The r e f l e c t i o n and expansion  v a l u e s o f the o r i g i n a l simplex d i d n o t always the b e s t response,  exceed  and f o r these s i t u a t i o n s a s e t o f  contraction operations existed.  These f o u r c o n t r a c t i o n  o p e r a t i o n s a r e presented, a l o n g w i t h c o n d i t i o n s o f usage i n Table V. A t t h i s p o i n t , i t becomes more f e a s i b l e t o look a t the a l g o r i t h m i n f l o w c h a r t form as presented i n F i g u r e 26. The  f l o w c h a r t w i l l i l l u s t r a t e the r e p e t i t i o u s nature o f  the r e f l e c t i o n , expansion and c o n t r a c t i o n o p e r a t i o n s , each o p e r a t i o n p r o v i d i n g an o p p o r t u n i t y t o produce a b e t t e r s e t o f f a c t o r s than the p r e v i o u s worst  set.  These  o p e r a t i o n s o s c i l l a t e back and f o r t h u n t i l the s o l u t i o n comes c l o s e t o the optimum where no s i g n i f i c a n t l y b e t t e r response  2.  can be o b t a i n e d .  The s t a r t i n g m a t r i x  A t the b e g i n n i n g o f t h i s d i s c u s s i o n , the f o r m a t i o n o f the o r i g i n a l simplex was i g n o r e d . procedure  advocated  by Spendlay  There was a s p e c i f i c  e t a l . (51) f o r the p r o d u c t i o n  of the s t a r t i n g m a t r i x , based on the number of v a r i a b l e s t o be o p t i m i z e d .  T h e i r method r e q u i r e d t h a t a range be  chosen f o r each o f the v a r i a b l e s and t h a t these v a r i a b l e s  115.  Table  V.  Summary  of the Simplex  Usage Condition  Operation/Formula  (1)  Reflection R  (2)  c  =  C +  Massive  contraction  0.50  (C -  F  (3)  mc  C  (4) '  w  C  mw  r  F  <  F  b  0.25  (C  = C -  0.50  worst  = C -  0.25  Resulting F-Ratio  F  rc  F  r c  F  r  <  F  r  F  mrc  -W)  contraction  Massive  >  reflection  = C +  Worst  n  Operations  W)  contraction R  Contraction  (C -  F  w  F  cw  W)  contraction (C -  <  W)  .  F  cw  <F  w  F  mew  116.  be : r e g u l a r i z e d to one.  The  by a l i n e a r t r a n s f o r m to range from zero  t r a n s f o r m i s as  follows:  x - R  min  y =  Eq.  57  dR  where  - any measured v a l u e w i t h i n the range  x  - regularized mm dR  value  - lowest v a l u e o f the range covered - the a b s o l u t e d i f f e r e n c e between the extremes of the range  The r e g u l a r i z a t i o n procedure  was  usually required  to n u l l i f y  d i f f e r e n c e s i n u n i t s and o r d e r s o f magnitude of the variables. response  Spendlay's r e g u l a r i z e d matrix from which the  simplex was  formed would be as f o l l o w s  for k  variables:  0  0  o  P  q  q  q  p  q  Eq.  58  117.  where: — -  kV2  q  '  + 1 }  Eq. 59  k + 1  =  The  { (k - 1) W k  1  Eq. 60  s t a r t i n g matrix presented a t the b e g i n n i n g o f  the simplex a l g o r i t h m d i s c u s s i o n was d e r i v e d f o r a 20,000 and 80,000 d a l t o n mixture, w i t h a d e f i n e d range o f 10,000 to 30,000 f o r the f i r s t component, and a range o f 60,000 to 90,00 0 d a l t o n s f o r the second component.  Spendlay's  r e g u l a r i z e d matrix i n t h a t case would be:  0.00  0.00  0.96  0.26  0. 28  0.96  Eq. 61  The v a l u e s i n the matrix were the s t a r t i n g  regularized  v a l u e s t h a t , i f converted back t o m o l e c u l a r weights by r e v e r s i n g the r e g u l a r i z a t i o n t r a n s f o r m produced  (Equation 57),  the s t a r t i n g m a t r i x shown a t the b e g i n n i n g o f  the d i s c u s s i o n .  118.  3.  The s i m p l e x  output  The  algorithm presented  simplex  was programmed  i n FORTRAN  (32), i n i t i a l l y  program t o a l l o w t h e t e s t i n g Through t e s t i n g  into  calculation  and t h e s i m p l e x MWD  iteration  c a n be s e e n  t h e o u t p u t was s i m i l a r  output.  The t i t l e  were p r i n t e d  s e a r c h was t o be p e r f o r m e d 90,000).  the molecular weight  routine,  t h e range pairs  and j u s t  the next  over which the • and 6 0 , 0 0 0  -  v a l u e s and  as t h e f i r s t  r e g r e s s i o n , demarcated by a l i n e  the f i r s t  best, worst,  three regressions, the actual  and next  by a p r i n t e d t o worst  procedure  of stars. simplex  iteration  F - r a t i o s were then  part  continued  value. then  printed.  The i t e r a t i o n  it  due t o no improvement b e i n g o b t a i n e d i n t h e  failed  -  (see E q u a t i o n 5 2 ) .  r e s u l t i n g F - r a t i o s were p r e s e n t e d  o p e r a t i o n began, s i g n i f i e d The  Iteration'  The FORTRAN p r o g r a m a u t o m a t i c a l l y g e n e r a t e d  regressed  After  t o t h e normal  (10,000 - 30,000  m a t r i x v a l u e s from  of  and t e s t e d  i n Figure 27.  'Simplex  t h e ranges  Spendlay's  The  r o u t i n e was  calculation  marked t h e b e g i n n i n g o f t h e i t e r a t i v e under t h e t i t l e  the routines  An example o f t h e b e g i n n i n g o f t h e  o f the simplex Initially  MWD  and e v a l u a t i o n o f t h e a l g o r i t h m .  t h e complete  "with model systems. output  as a s e p a r a t e  a g a i n s t manual c a l c u l a t i o n s ,  were shown t o be w o r k i n g , incorporated  i n F i g u r e 26  until  Figure  26, Flowchart for the simplex algorithm. Spendlay's matrix  ** y N  yes no  Go to 1  Centrdid (CR)  Centroid  Massive Reflection Contraction  F•re > Fr' ?  Reflect  Go to  Regression  Regression  Centroid (E)  Reflection Contraction  F  Fw > Fr < Fn  Fn > F_r < P.b  F  mcr > * r  2  1  End  I  r  > F  expand  b ?  Regression  Fe > Fr ?  Centroid (MCR)  Go to  Centroid (R) Go to 1  1  Fr < Fw _?  Centroid Go to  1  (MWC)  Worst Contraction  Fmwc > Fw ? Contraction \Failure - End  Regression  Regression  Fcw > Fw ?  Massive Worst Contraction  Centroid (CW)  Go to  1  C M 3. xx'>> > »">') >>>>>>< ifrrrrr/>>t)>r .. WARM._.. 1 . . Uif-b. ^ . 2.WARM . . 3 .  ' r r x r X X xx x X x x x x x X xx X X X X XX. ^« XXXX X X X X X X X x x x x x x x x x ~ -". X X X X X / \ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ARM ' ". . .~4 . ".W ARM , i> . - A R M . . .b . . . . 7 . . W A R M . . . 8 . . WARM. . 9 . . WAR/-'. . . 0 . . W A R M . . . 1 . . W A R M . . . 2 . . W A F. M . • . 3 . i'5:"ll":69 F R I APR 2 2 / 7 7 " Ml S l / . R l i ' 7) U M V E R S I TY C F K C L C M i u T I . ' . G ILMi-.tr' x  RI s !-.). i V j l i s ThE  CCfFLTINC  CICSEC  ALL C A r S A T . APRIL  23 -  RENOVATIONS * *  rOMN  J S I G  Figure 27.  TTTTTTTTTTTT TMTTTTITTTT  T 1 T T T T  rr rr  TT I r TT  **LAST LSER  cctccccccccc  CCCCCCCCCCCC'"  cc cc cc cc cc  cc  SICNCN "TCH-"  M,MDER  M.. ats. v  "NC.-EER"  M MM  MMM  of the simplex  MM MMM  '  CC  W A S : 15 : I C : 16 SICNEC  CN AT " 1 5 : 1 1 : 1 0  CciJ.SSCCFAM-Cej.SIMl.C2  EAECUTICN  MM  I'M (•MM »M<h  Initial output " a l g o r i t h m . ' '"  MMVM yyyy MM M M yy yy yy ty yy yy MM MM ~ c c > M ~ "MM MM"" ;•>• yy MMMM . MM MM MM. cc yy yy ^ ^ MM MM cc 1" M yy MMMM CM MM CC " cc t-yM.V MM cc cc yy yy MM cc cr. r M yy y y MM MM CCCCCCl'CC ecu"" yy MM MM CM MM CCCCCCCCCCCC  TT TI  iXL.S  C ENT FR U l L ' B .  .%M.kH  C N FR I  APP 2 2 / 7 7  1 b = — K 1JMA7  b=C!NS  OF MCLECULAF  WEICFIS  OF C A T A C ; ? C S C - F ' C Y C  C  E  S  2 "  1  25 " "  MCLcCULA.' ..JiCFT IMERVJL 1.7C CALCULATION MCUE 1 n F c R t : 3 = M W L C A L C U L A T I O N MODE " " 1= I T T E R A T I C N  MCC6  ' 2»SMCOTHNING  MODE  r  —  -  "  -  "  t  "  "  O ; The  FeRTINANT  P AR A M ; TF. F S F C K T H I S  CAICIIAIICK  AFE:  j •  CcLL  D C T TCV  7.2CCCO  MF.MSCUS  •  6.80UOO  '  .  ;  '  J  121.  co m -a* •m  'n o m p a * [O O r\i  Si.  a — a;  oo • boo !oo  i  '  ;  0  J! - UV c*  —» .n Ul o m m ,CM • c» •O rn m o n UJ UJ) ~« m rn o ru T r> g ••**  a<: Ux Oi> | na s  jy» vr> j"* I• j I in Ul >n ' . * >  eg —• o  —» ut C» m ci .o m m CJ o LU UJ m u\ •o rg r* rg m  u <i m n V* Ul it Ul \* t/> ro O •n en r<i j CJ o 1 UJ 1 A rM AJ '•o«-» U.Jo  o r» cj ut m CM p O ci 1 m m ) 1o !~«  IJJ  1 1  «># W — r\• J J U•Jo •a •o U  sotsft  S9IDN3n33HJ 3 Hi  •itit  sstaa's  -SI  S3IH3S  S9S*tIS CN CN H  .  . .  .  ._  *H 3*1 IS31-J  5  «33VTJ W3UVH3M1  «**«*«**t*»*«»«««*»««*************»*a«***«*ft**»*4**«j»4*******«a*«4*«4«««**»•*** »t.M*J I wm»HttMM.*lJ-tMKt 111 l » M M_. 66VOSM  093i3'D  •i8/9E  VHVJ  -3«tf S3IDN3n03>H 3Wi -SI  S3ld3S *W 3H1 1S31-3  EDS'U ? . 1X3M  '0<rL.\  ISH3V  *8/9  d33wriN * 3 U V _ l l l  ..... »  '81*2  1S3J-J  EDS* a ?  6IS»E'I  E&St 3*0  '8*583  • ?115 T  iS3B  -3a?  5313N3nri3«d__i_  -SI  S3IS3S "W 341 1S31-3  *VO"3I*2  »**««»*«*»*»»*«»»««•**«**»*****•***•*»*****«*«*********»»**•«*•**»**********•****»**•*••***•**•«•*****»*.***********•*••**••** ' .  '  "  "  J  L  E  i  9  i  f  3 _?_L9.  -  3  H  S3I3N3n;>3_ 3-11  »  _U__3.Ld__l__Wi_  __.!_!  1S31-J  _ 355*>CL I _*.***tt*****4*********1>*.****.*.*.^**.**.*J*^  (-*»_•.•*-•.».•*>.«__•.*-• *•**.»*.•.•.•».•.»•..  •00006  ************************** 3NIinf)>l HTH1VM31I X33dHIS  ViEVc'l  e/ZZ9*3  •00009  •0DD3I  •00009  'ODOOE  -3M? S3IDN3nft3>H 3H1 -SI '00D3I  S3IW3S CH 3'U CV  c  123.  F-test  o r due  being  t o t h e maximum l i m i t  exceeded.  The  results  s e t on  of t h i s  the  iterations  particular  example  were:  M  1  =  24,400  M  2  =  80,000  .93 f  =  2  T h e s e r e s u l t s were i n e x c e l l e n t  .92  agreement w i t h t h e  m o d e l o f a 25,000 - 80,000 d a l t o n m i x t u r e , ratio.  E.  of  presented  data  Although  solution  the i t e r a t i v e  the c o r r e c t molecular weight,  weight,  and  was  drawback.  original undefined  and  information T h i s was  the r a t i o s  complete  distribution  was  o b t a i n e d a t one  c o u l d , however, be from  done by  thereby  MWD  MWD  the  iteration  inserting  there  not provided. was  The  usually-  A representative combining  the o r i g i n a l  a dummy s t a t e m e n t  to reinforce  molecular -  components,  o b t a i n e d by and  simplex  the e s t i m a t i o n  average  speed  1:1  on.  allowed  of the i n d i v i d u a l  A complete  of the  later  weight  not very presentable.  regression matrix and  be  S m o o t h i n g o f 'undefined  one  present i n  F u r t h e r examples o f the c a p a b i l i t i e s  optimization routine w i l l  original  to correct  smoothing the d i s t r i b u t i o n .  distribution.  into  the  frequency This  the  values,  technique  124.  was  very  similar  t o t h a t used  i n t h e removal o f t h e  intercept  from  statement  and where i t w o u l d a p p e a r i n t h e r e g r e s s i o n  matrix  i s presented  c(5)/c  where  the regression.  An example o f a dummy  below.  ln  0  K  2n  K  4n  0.5  1.0  1.0  3n 0.0  0.5  0.0  0.0  1.0  1.0  0.0  0.0  0.0  0.0.  •0.0  0.0  a - end o f t h e r e g r e s s i o n m a t r i x b - smoothing s e t c - null set  T h i s dummy s t a t e m e n t the  was f o r a two component c a s e  sum e q u a l l e d 1.0.  The v a l u e s o f 1.0 a r e a d d i t i o n a l  dummy k^_. v a l u e s w h i c h e m p h a s i z e t h e m o l e c u l a r contributing statement via  t o e a c h component.  illustrated  a special  the i t e r a t i o n calculation  c o u l d be e n t e r e d  procedure.  Performing  including  into  the matrix  obtained  a complete  from  MWD  t h e s e dummy v a r i a b l e s  t o t h e smoothing o f t h e o r i g i n a l u n d e f i n e d  led  distribution.  Examples o f t h e smoothing c a p a b i l i t y o f t h i s c a n be s e e n  weights  The t y p e o f dummy  subroutine using the r a t i o s  o f data  where  procedure  i n F i g u r e s 2 8 and 29, t h e smoothed v e r s i o n s  o f F i g u r e s 19 a n d 20.  125.  1.0  0.6  f  0.2  14  28  56  112  224  -3 M x 10 Figure 28.  Semilcgarittimic p l o t o f f (M) vs. M f o r a 25,00080,000 dalton irdxture (1:1 r a t i o ) . Single speed data; i n t e r v a l -- 2.0, with data having undergone smoothing.  0.3  0.2  Y  0.1  0.0 640 M x 10 ?  i g u r e 29.  SemllcjgafltiTmic p l o t o f f (M) vs. M f o r a 25,00080,000-320,000 dalton mixture (1:1:1 r a t i o ) . Single speed data; i n t e r v a l = 2 . 0 , with data having undergone siroDtljing.  126.  Case S t u d i e s o f Some Model Systems  In order t o i l l u s t r a t e the p o t e n t i a l o f the MWD c a l c u l a t i o n w i t h r e s p e c t t o p r o t e i n systems, four case s t u d i e s o f p r o t e i n s w i l l be presented. represented  The chosen systems  s i t u a t i o n s t h a t c o u l d a r i s e d u r i n g an u l t r a -  c e n t r i f u g a l i n v e s t i g a t i o n of p r o t e i n s .  The systems  i n c l u d e d a d i s c r e t e s e l f - a s s o c i a t i n g system, a three component r v x t u r e , and two c o n t r a s t i n g i n t e r a c t i n g systems.  A l l were generated u s i n g t h e Rinde  equation  model and had t h e same p a r t i a l s p e c i f i c volume and extinction coefficient. m u l t i p l e speed data  The models were analysed  (three speeds) and unispeed  The weight average molecular  51.  The simplex  data.  weights o f the components  were c a l c u l a t e d f o r some o f the d i s t r i b u t i o n s Equation  using  using  i t e r a t i v e r o u t i n e was used t o  o b t a i n the b e s t f i t weight average m o l e c u l a r  weight,  and hence the component c o n c e n t r a t i o n , i n o r d e r f o r comparison t o the o r i g i n a l model.  A.  Catalase  (57,000  Catalase  i s known t o s e l f - a s s o c i a t e t o form a  tetramer. and  -  232,000  daltons;  1:2  ratio)  T h i s type o f i n t e r a c t i o n i s common t o p r o t e i n s  i s o f t e n a f f e c t e d by environmental f a c t o r s , such as  127.  pH  and i o n i c  in  detecting the extent  the  strength.  tetramer  monomer.  was  molecular  M  These  of  the following  ±  data,  the smoothing  d e s i r a b l e form  the weight  =  .25  =  .75  presented  average  of the  using  model.  being  This  somewhat  could  which  be  weight  31. using  was  undefined.  subjected  i t changed  i n Figure  molecular  the  b u t were n o t v e r y  the d i s t r i b u t i o n  routine, after  produced  itself,  t o the o r i g i n a l  due t o t h e d i s t r i b u t i o n  t h e same  study,  results:  were n o t u n r e a s o n a b l e , compared  useful  case  Assessment  the d i s t r i b u t i o n  f  when  In t h i s  data  30.  = .253,000  partially  more  from  could be  the concentration of the  i n Figure  f  results  Using  i n twice  55,800  accurate  to  presented  51, p r o d u c e d  =  2  of interaction.  present  weight  Equation  1  calculation  Analysis of the multispeed  distribution  M  T h e MWD  t o t h e much  Calculation Equation  51.  produced:  M  ±  M  The  2  =  61,600  f  =  231,000  f  application  original  data  ±  of the simplex  produced  2  = .31 =  .68  iteration  routine to the  the following values:  128.  /  0  \  o ' O-  \  o  O  15  ,  \  i\  1  30  60  O  V  /  0-J  I  120 M x 10~  Fig\ore 30.  , o  240  I  480  \  960  3  S e m l o g a r i t l i m i c p l o t o f f (M) v s . M f o r a 57,000232,000 d a l t o n m i x t u r e (1:2 r a t i o ) . 5,000, 10,000 and 15,000 rpm, i n t e r v a l = 2.0.  M x 10 F i g u r e 31.  SeTmlcigaritljmic p l o t o f f (M) v s . M f o r a 57,000232,000 d a l t o n m i x t u r e (1:2 r a t i o ) . 5,000, 10,000 and 15,000 rpm, i n t e r v a l = 2.0, w i t h d a t a h a v i n g undergone smootliing.  =  57,000  f  = 232,000  f  = .33  ±  2  = .66  These r e s u l t s were i n e x c e l l e n t agreement w i t h the o r i g i n a l model.  A n a l y s i s of s i n g l e  distribution  speed d a t a produced the  i l l u s t r a t e d i n F i g u r e 32, a b e t t e r d i s t r i b u t i o n  than f o r the o r i g i n a l t h r e e speed d a t a .  This  situation  arose i n some cases, i l l u s t r a t i n g how the c h o i c e o f speed c o u l d s i g n i f i c a n t l y a f f e c t the d i s t r i b u t i o n .  Calculation  of the weight average m o l e c u l a r weight u s i n g E q u a t i o n 51 produced:  M M  1  =  52,300  f  2  = 237,000  f  = .28  ±  2  = .72  A p p l y i n g the simplex o p t i m i z a t i o n r o u t i n e t o unispeed data, produced  the f o l l o w i n g :  1  56,700  f  1  = .33  2  = 232,000  f  2  = .66  M ..= M  Again t h i s was i n e x c e l l e n t agreement w i t h t h e o r i g i n a l model.  130.  B.  T r y p s i n i n h i b i t o r - ovalbumin - conalbumin (17,000-45,000-85,000;  1:1:1  ratio)  T h i s case r e p r e s e n t s a mixture r e l a t i v e l y c l o s e together interactions  of three proteins  i n m o l e c u l a r weight w i t h no  taking place.  The f i r s t a n a l y s i s o f t h r e e  speed d a t a was made u s i n g a m o l e c u l a r weight i n t e r v a l o f 2.0, w i t h the r e s u l t i n g F i g u r e 33.  d i s t r i b u t i o n presented i n  C n l y two components c o u l d be detected,  due t o the poor r e s o l u t i o n large i n t e r v a l .  resulting  mainly  from the r e l a t i v e l y  By changing the i n t e r v a l t o 1.5, the  d i s t r i b u t i o n changed, as i l l u s t r a t e d i n F i g u r e 3.4. Three components were then e v i d e n t , although became somewhat undefined.  Further manipulation  i n t e r v a l and range would enable d i s t r i b u t i o n t o be o b t a i n e d . r e q u i r e d s i n c e the simplex solution.  a defined three  o f the component  T h i s o p e r a t i o n was not  o p t i m i z a t i o n c o u l d p r o v i d e the  The r e s u l t s o f the simplex  o p t i m i z a t i o n were:  M  1  =  16,800  f  M  2  =  43,000  f„ = .32  M  3  =  84,000  f  S i n g l e speed simplex  the d i s t r i b u t i o n  x  3  = .32  = .35  o p t i m i z a t i o n produced:  131.  0.8  O  0.6  *  o S 0.4 O-o  o  0.2  0.0  15  30  60  _G—1_ 120 -3 M x 10  240  480  960  Figure 32. SeMlogari+JnTtdc plot of f (M) vs. M for a 57,000232,000 dalton mixture (1:2 ratio). 13,000 rpm, interval =2.0.  0.6  0.4  0.2  h  0.0 M x 10 Figure 33. Semilogarithmic plot of f (M) vs. M for a 17,00045,000-85,000 dalton mixture TT:1:1 ratio). 10,000, 20,000 and 30,000 rpm, interval = 2.0.  132.  M  1  =  16,400  M  2  =  42,200  =  84,600  M  3  f, = .31  F i g u r e 35 shows a smoothed f u n c t i o n u s i n g the r e s u l t s o b t a i n e d from the s i n g l e speed i t e r a t i o n .  The smoothing  r o u t i n e d i d not, i n t h i s case, produce e x a c t l y the d i s t r i b u t i o n d e s i r e d , p a r t i a l l y due t o the e m p i r i c a l n a t u r e o f the c a l c u l a t i o n .  The smoothing  r e q u i r e d t h a t the r e g i o n o f b e s t f i t  routine  frequency response  be strengthened, which was not always p o s s i b l e when the m o l e c u l a r weights were so c l o s e l y  C.  spaced.  . a s^ - c a s e i n and K -casein, and i n t e r a c t i o n , p r o d u c t (25,000-400,000-1,500,000;  2:2:1 r a t i o )  T h i s i n t e r a c t i n g system r e p r e s e n t s one o f the more difficult  systems s t u d i e d .  Both °t.s^ and K - c a s e i n a r e low  m o l e c u l a r weight components, but K - c a s e i n aggregates t o a v e r y h i g h m o l e c u l a r weight.  When a s ^ - c a s e i n was  i n t r o d u c e d i n t o the system, an i n t e r a c t i o n p r o d u c t o f i n t e r m e d i a t e m o l e c u l a r weight was formed.  Analysis of  t h i s system a t t h r e e speeds produced the d i s t r i b u t i o n illustrated  i n F i g u r e 36.  In the c a l c u l a t i o n s , t h e  m o l e c u l a r weight i n t e r v a l had been i n c r e a s e d t o 2.2, t o  133.  n.50 O  O  o &  o  0.25!  \  o  O  o  \  Q i  o  0.00  \ j  \  -Cl-  n  10  22 Mx  F i g u r e 34.  O  t -3 10  113  50  S e m i l o g a r i t J m L c p l o t o f f (M) v s . M f o r a 17,00045,000-85,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 10,000, 20,000 and 30,000 rpm, i n t e r v a l = 1.5.  0.50  O N  o  s ^  O..  0.25  -O  o  o  o  o  o /  o  J  10  o  -OL  22  50 M x 10  F i g u r e 35.  o  \  o o  0.00  o  110  -3  S e i i i i l o g a r i t h m i c p l o t o f f (M) v s . M f o r a 17,00045,000-85,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 25,000 rpm, i n t e r v a l = 1 . 5 , w i t h d a t a h a v i n g undergone smoothing.  134.  accommodate the wide range o f m o l e c u l a r weights.  By  a p p l y i n g the simplex o p t i m i z a t i o n r o u t i n e t o m u l t i p l e speed d a t a , the f o l l o w i n g r e s u l t s were o b t a i n e d :  M  x  =  24,000  f  M  2  =  399*, 000  f  2  = .39  M  3  = 1,491,000  f  3  = .19  = .39  1  The d i s t r i b u t i o n o b t a i n e d from a s i n g l e speed can be seen i n F i g u r e 37.  Simplex i t e r a t i o n f o r the s i n g l e  speed d a t a produced:  M  D.  1  =  24,000  f  = .39  M  2  =  402,000  f  2  = .39  M  3  = 1,550,000  f  3  = .21  1  Lysozyme - ovalbumin and i n t e r a c t i o n product (17,000-45,000-62,000;  2:2:1 r a t i o )  T h i s study was o f an i n t e r a c t i o n a t the o t h e r extreme of the r e s o l u t i o n s c a l e . product d i f f e r e d l i t t l e ovalbumin.  In t h i s case, the i n t e r a c t i o n from the m o l e c u l a r weight o f  A n a l y s i s o f m u l t i s p e e d d a t a produced a two  peak d i s t r i b u t i o n .  Simplex o p t i m i z a t i o n f o r t h r e e  components generated the f o l l o w i n g  results: 1  135.  M  = 17,000  f  M  2  = 45,800  f  M  0  = 63,000  3  '  = -40  ±  = .41  2  f , = -18 J  A n a l y s i s o f s i n g l e speed data u s i n g the simplex o p t i m i z a t i o n produced  M  the f o l l o w i n g r e s u l t s :  = 16,900  f  M_ ••= 42,000  f  M  f  x  3  = 59,300  x  ?  3  = -39 = .31 = -29  In t h i s case, the simplex c a l c u l a t i o n was n o t a b l e t o y i e l d r e s u l t s as  accurate  as i n p r e v i o u s systems.  This  c o u l d have been a speed r e l a t e d problem, where the s i n g l e speed d i d not c a r r y enough i n f o r m a t i o n t o l o c a t e t h e c o r r e c t optimum.  Discussion  In a l l cases, the three speed d a t a produced reasonable d i s t r i b u t i o n s .  A l l of these  fairly  distributions  p r o v i d e d an i n d i c a t i o n o f the number o f components p r e s e n t , and o f the p a r t i c u l a r molecular weight r e g i o n s i n which the simplex would begin i t s s e a r c h .  The simplex  o p t i m i z a t i o n worked v e r y w e l l , w i t h a l l m u l t i s p e e d  cases  g i v i n g a c c u r a t e r e s u l t s t h a t r e f l e c t e d the r a t i o s and m o l e c u l a r weights  o f the o r i g i n a l models.  From t h i s  136.  i n v e s t i g a t i o n i t was apparent t h a t i f t h e c e n t r i f u g a l data approached the accuracy  o f the model data, the simplex  o p t i m i z a t i o n procedure c o u l d r e a d i l y s o l v e f o r r e l a t i v e concentrations,  i n a wide v a r i e t y o f two and t h r e e  component systems. In the case o f s i n g l e speed data almost the same c o n c l u s i o n s c o u l d be made, except i n the r a t h e r case o f the lysozyme-ovalbumin i n t e r a c t i o n . . simplex  difficult  Use o f the  o p t i m i z a t i o n removed t o some extent the r e q u i r e -  ment t h a t a d e f i n e d d i s t r i b u t i o n be o b t a i n e d , a good i n d i c a t i o n was p r o v i d e d components p r e s e n t  as t o the number o f  i n a g i v e n system.  c o u l d come from e i t h e r the MWD  as long as  This  indication  or from p r e v i o u s  knowledge  of the components i n the mixture.  Analysis of Protein  Mixtures  Up t o t h i s p o i n t o n l y model systems have been  analysed.  F o l l o w i n g the development and v e r i f i c a t i o n o f the computer programs, and the f i n a l removal o f a l l the problems, some u l t r a c e n t r i f u g e runs were completed. w i t h standard at  These runs were made  p r o t e i n s d i s s o l v e d i n 0 . 0 5 M phosphate b u f f e r  pH 7, d i a l y s e d f o r 24 hours, d i l u t e d t o an absorbance  of 0.5 and then mixed i n a 1:1 r a t i o by volume. experimental  The  procedures d e s c r i b e d f o r u l t r a c e n t r i f u g a t i o n  i n P a r t I were c a r r i e d o u t f o r these  runs.  137.  0.4  M x 10 F i g u r e 36.  S e r M l o g a r i t h m i c p l o t o f f (M) v s . M f o r a 25,000400,000-1,500,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 3,000, 7,000 and 13,000 rpm, i n t e r v a l = 2.2.  0.4  O /\  O / i / /  0.3  i  G  >  I  S  0.2  /  ,  /  -  ' G  1  1  1  o  o \  o I  1  1  in  0.0 6  29  in  i  Q  n\  140 M x 10  F i g u r e 37.  i  O  1  /' '  i  O  0  O  i  '  I 1  ; v  i  °\\  i 680  1 rt>  / V  1  l 1 I  /  1 1  /  G \  » \ »  » I  i  l  i  \  O  i  3292  0\  3  Serrdlogarithmic p l o t o f f (M) v s . M f o r a 25,000400,000-1,500,000 d a l t o n m i x t u r e (1:1:1 r a t i o ) . 10,000 rpm, i n t e r v a l = 2.2.  138.  Numerous problems were encountered c e n t r i f u g e d u r i n g our attempts mixtures. was  w i t h the  t o o b t a i n data f o r p r o t e i n  The d i f f i c u l t y i n o b t a i n i n g a c o r r e c t b a s e l i n e  one of the most s e r i o u s problems.  What e f f e c t  i n c o r r e c t b a s e l i n e c o u l d have on the MWD  i s unknown.  D i f f e r e n c e s i n p a r t i a l s p e c i f i c volume and c o e f f i c i e n t s among p r o t e i n s caused  an  extinction  further complications  i n experimenting w i t h r e a l systems.  Since the program  c o u l d not handle i n d i v i d u a l p a r t i a l s p e c i f i c volumes, an average v a l u e was The  used.  f o l l o w i n g experimental r e s u l t s r e p r e s e n t some of  the more s u c c e s s f u l runs.  A l l the samples were run a t  m u l t i p l e speeds w i t h the analyses b e i n g performed on t h r e e speed  A.  data.  T r y p s i n i n h ib1tor-conalbumin  (1:1 by volume)  The r e s u l t s of t h i s experiment The mixture was 19,600 rpm. MWD  mostly  run a t three speeds:  The  C(^)/C  q  were v e r y  encouraging.  9,500, 14,400 and  data were run through the complete  c a l c u l a t i o n and the r e s u l t i n g d i s t r i b u t i o n can be  i n F i g u r e 38.  A w e l l d e f i n e d d i s t r i b u t i o n was  w i t h the weight average molecular weights E g u a t i o n 51 as f o l l o w s :  seen  obtained  determined  by  139.  The  M  1  =  23,000  f  1  =  .52  M  2  =  82,000  f  2  =  .49  data  produce When  were  then  put through  the distribution  t h e raw d a t a  routine,  illustrated  f i t solution  found  was  1  =  ?4,000  f  M  2  =  98,000  f  rpm)  produced  ±  obtained  of these  routine,  produced:  M  only  the distribution  Analysis  M  data,  using  2  =  -51  =  .47  from  the simplex  =  12,700  f  2  =  64,800  f  1  2  =  .28  =  .63  39. optimization  t o be:  a single  presented  ±  'Trypsin  i n Figure  the simplex  M  of data  routine to  were p u t through  the best  Analysis  B.  the smoothing  speed  (19,620  i n Figure  40.  optimization  inhibitor-ovalbumih-conalbumin  (1:1:1 b y volume)  Analysis  of three  21,600  rpm)  produced  Figure  41.  Simplex  three  speed  data  (9,600,  the distribution  and  illustrated i n  optimization of this  components generated  15,500  data  setfor  t h e f o l l o w i n g optimum  values:  140.  a. 8  a.6  S  r  0.4  0.2  0.0 14  28  112  56  224  M x 10  Figure 3 8 .  Semilcgaritlimic plot of f(M) vs. M for a mixture of trypsin inhibitor and conalburrxLn. Absorbance = 0 . 5 for both, mixed . 1 : 1 by volume. Run at 9,600> 15,500 and 2 1 , 6 0 0  rpm.  0.8  JO, 6  g  0.4  '  o o  3-2 o  u-0  Q 14  28  56  112  224  -3 M x 10  Figure 3 9 .  Semilogarithmic plot of f (M) vs. M for a mixture of trypsin inhibitor and conalbumin. Absorbance = 0 . 5 for both, mixed 1 : 1 by volume. Run at 9 , 6 0 0 , 15,500 and 21,600 rpm, with data having undergone smoothing.  141.  0.8  o  -  0.6  0  >  \  1  o  S  €  0  1  0.4  /  \  o  1  ,  1 1 \  - o /  o  O  0 \  1  O  i  \  1  \  14  1  \  / 1  \  /  0.0  \  1  1  0.2  \  \  O  I  I  28  112  56  M x IO  224  - 3  Figure 40. Serailogarithmic p l o t o f f (M) v s . M f o r a mixture of t r y p s i n i n h i b i t o r and conaJ^urrdn. Absorbance = 0 . 5 f o r both, mixed 1:1 by volume. Run a t 19,600 rpm.  0.50 O „ 0  o'  1  i  LLT  I'  0.25  , i  O  7  '  \  /  i I  ?  \  1  x  \  /  1  i 14  n  o  i n 28  O  \  i  -3  l  1  56  112  \  r> 224  M x 10 Figure 41. SemLlcigarittimic p l o t o f f (M) ' vs. M f o r a mixture of t r y p s i n i n h i b i t o r , ovalbumin and conalburtiin. Absorbance = 0.5 f o r a l l , iriixed 1:1:1 by volume. Run a t 9,600, 15,500 and 21,600 rpm.  14 2.  C.  M  ±  =  10,000  f  M  2  =  49,128  f  2  = .43  M  3  = 115,000  f  3  = .22  Ovalbumin-thyroglobulin  = .37  ±  (1:1 by volume)  T h i s system i n r e a l i t y r e p r e s e n t s a t h r e e component distribution  s i n c e t h y r o g l o b u l i n i s made up o f two s u b u n i t s .  The d i s t r i b u t i o n o b t a i n e d from t h r e e speeds  (6,503, 9,620  and 15,526 rpm) i s presented i n F i g u r e 42, which shows the presence o f three components.  Simplex i t e r a t i o n o f  the data produced the f o l l o w i n g r e s u l t s :  D.  M  x  =  37,000  ±  M  2  =  388,000  f  2  = .17  M  3  = 1,040,000  f  3  = .21  1  = .43  Catalase  C a t a l a s e i s a two component system s i n c e i t e x i s t s i n two forms, a monomer and a tetramer.  T h i s was  demonstrated by the d i s t r i b u t i o n , even though i t was undefined, obtained a t t h r e e speeds and p r e s e n t e d i n F i g u r e 43.  Simplex i t e r a t i o n o f t h e data produced:  143.  0.50 G /\ G/ G  O l  i  G  20  \  \  o  •  o  >  o  o  7  G o'  \  o —LO  N  Ci  1  80  O *  1  320 M x 10~  1  ,  l_  o_  1280  J  Figure 42. Semilcgarithmic plot of f (M) vs. M for a mixture of ovalbunin and thyroglobulin. Absorbance = 0.5 for both, mixed 1:1 by volume. Run a t 6,300/ 9,600 and 15,500 rpm.  320 M x 10 Figure 4.3.  1280  3  Semilcgarittimic plot of f (M) vs. M for catalase. Absorbance = 0.5, run at 9,600,15,500 and 21,600.  144.  M  E.  54,800  40  M - = 379,000  60  a  =  Discussion  Although  the experiments  performed  were n o t as  c o n c l u s i v e as d e s i r e d , they d i d i l l u s t r a t e c a l c u l a t i o n c o u l d e x t r a c t a MWD though the molecular weights  that the  from t h e system.  molecular  and r a t i o s t h a t w e r e w i t h i n r e a s o n a b l e  w i t h t h e i r known v a l u e s .  Even  a n d f r e q u e n c i e s c o u l d n o t be  c o n s i d e r e d c o r r e c t , each d i s t r i b u t i o n produced weights  comparison  In order f o r experimental  s i t u a t i o n s t o be e v a l u a t e d p r o p e r l y , more a c c u r a t e w o u l d be r e q u i r e d , as w e l l as d a t a from a t l e a s t or three  MWD  data  two  speeds.  S i n g l e speed d a t a h a s n o t been g i v e n any p r o m i n e n c e i n the a n a l y s i s of the experiments.  T h i s was m a i n l y  t o t h e c o n f l i c t g e n e r a l l y e x i s t i n g between s i n g l e and  due  speed  t h r e e s p e e d r e s u l t s , a s shown i n t h e c a s e o f t r y p s i n  inhibitor-conalbumin.  The r e a s o n  f o r t h i s was n o t known,  b u t i t was p o s s i b l e t h a t t h e s i n g l e s p e e d s c h o s e n f o r a n a l y s i s were n o t t h e most a p p r o p r i a t e . t h e known c o m p o s i t i o n o f t h e s y s t e m s ,  Considering  the multispeed  w e r e more r e p r e s e n t a t i v e t h a n m o s t o f t h e s i n g l e results.  data  speed  145.  A f u r t h e r f a c t o r t o be c o n s i d e r e d was the u l t r a centrifuge i t s e l f .  Although  the p r e p a r a t i v e u l t r a c e n t r i f u g e  was u s e f u l f o r the r o u t i n e assessment o f m o l e c u l a r  weights,  i t was q u e s t i o n a b l e whether the data o b t a i n e d were s u f f i c i e n t l y a c c u r a t e f o r the d e t e r m i n a t i o n o f m o l e c u l a r weight d i s t r i b u t i o n s .  D i s c u s s i o n s w i t h o t h e r workers (40)  r e v e a l e d some concerns whether the p r e p a r a t i v e u l t r a c e n t r i f u g e c o u l d be used f o r t h i s r e l a t i v e l y s o p h i s t i c a t e d analysis.  In a d d i t i o n , UV o p t i c s l a c k e d "the i n h e r e n t  p r e c i s i o n o f s c h l i e r e n o r RayleLgh i n t e r f e r e n c e o p t i c s " (12). O b v i o u s l y , t h e b e s t method f o r t h e assessment o f e x p e r i m e n t a l d a t a would be t h e use o f a Model E u l t r a c e n t r i f u g e with Raleigh interference o p t i c s . c a p a b i l i t i e s o f the MWD  equipped  E v a l u a t i n g the  c a l c u l a t i o n using a less sophisticated  instrument may l e a v e some doubt as t o t h e v a l i d i t y o f the results.  T h e r e f o r e , the experiments performed can o n l y be  c o n s i d e r e d t o p r o v i d e an i n d i c a t i o n t h a t the MWD c o u l d be o b t a i n e d . of  of proteins  E v a l u a t i o n o f the a c t u a l c a p a b i l i t i e s  the c a l c u l a t i o n w i l l have t o be l e f t t o groups having  access t o a Model E u l t r a c e n t r i f u g e and a d e t a i l e d knowledge of  the p r o t e i n systems t o be assessed.  Other F a c t o r s  A.  Extinction  coefficients  146.  The  d e r i v a t i o n o f the Rinde e q u a t i o n  assumed  e q u i v a l e n t e x t i n c t i o n c o e f f i c i e n t s so t h a t t h e e q u a t i o n c o u l d be w r i t t e n i n terms o f c o n c e n t r a t i o n ambiguity.  without  I t i s w e l l known t h a t t h i s i s n o t the case,  s i n c e e x t i n c t i o n c o e f f i c i e n t s o f p r o t e i n s can v a r y markedly.  To account f o r v a r i e d e x t i n c t i o n c o e f f i c i e n t s ,  the f o r m u l a t i o n had t o be changed.  Absorbance i s r e l a t e d  to c o n c e n t r a t i o n by 3eer's law:  A = abc  where:  A - absorbance a - absorptivity b - l e n g t h o f the l i g h t path (cm) c - . c o n c e n t r a t i o n (g/1)  Reformulating  A  the Rinde equation  ( £ )  \ - >  =  X.M.  i n these  exp(-X.M.C)  — exp(-A.M.) 1 - exp(-A .M. ) L  A_  terms g i v e s :  L  f  i  E<  2-  6  4  1  where;  A f  =  Q  . 1  A_ o  _  a.c . 1 oi A o  Eq. 65  14 7.  Expanding the summation and r e d u c i n g the bulk term t o k produces: A (£) A  a.c. -=-°i K, A o  = o  By examining  +  a.c. -=-°i K„ . . . + A o  a c . K A o  -3-Sa  Eq.  66  the above equation i t can be seen t h a t the  c o n t r i b u t i o n o f each component i t s extinction coefficient.  i s directly  related.to  A component w i t h twice the  a b s o r p t i v i t y of another would a c t as i f i t was  present  i n twice the amount, and t h e r e f o r e have twice the area i n the c a l c u l a t e d MWD.  In the case of w e l l r e s o l v e d  d i s t r i b u t i o n s of d i s c r e t e p r o t e i n components, a simple area adjustment  should s u f f i c e t o c o r r e c t the d i f f e r e n c e .  T h i s was shown to be the case when e q u i l i b r i u m p a t t e r n s of two p r o t e i n s , having d i f f e r e n t m o l e c u l a r weights  and  a b s o r p t i v i t i e s , were s y n t h e s i z e d , summed and t h e i r d i s t r i b u t i o n s analyzed.  Through simple d i v i s i o n by a  c o n s t a n t r e l a t i n g the a b s o r p t i v i t i e s , the d i s t r i b u t i o n c o u l d be c o r r e c t e d .  T h i s c o r r e c t i o n was o n l y u s e f u l when  the components were completely separated, o r the r e g i o n s of t h e i r d i s t r i b u t i o n s w e l l d e f i n e d .  B.  P a r t i a l s p e c i f i c volume  D i f f e r e n c e s i n the p a r t i a l s p e c i f i c volume o f p r o t e i n s have been ignored up to t h i s p o i n t i n the d i s s e r t a t i o n .  148.  Any  difference  different MWD.  components making  I n t h e program  multiple  regression  differences could was  i n the partial  calculation  the  correct  that  higher  daltons  partial  This program however,  volume  that  was  found  used  i n the  MWD  be  matrix,  i n effect.  volume  a a tetramer  than the volume  o f 10,000  using  of the  t o 80,000,  of the tetramer  t o the.  i nthe  the correct  component.  i n the original  t o work w e l l .  o f the system  used f o r  t o form  specific  be c a l c u l a t e d  used  was  a l l t h e K_^_. t e r m s  f o r each  The  by an example:  region  volume  value  be  of the regression  specific  proper use of t h i s  knowledge  thesis,  could  o f the range  the p a r t i a l  specific  approach  a n d was  component  volume would  partial  matrix could  specific  and i n the  o f t h e components  self-associated  v a l u e s o f the range,  regression  prior  region  t o a molecular weight  the p a r t i a l  the  n o t be t h e case.  be e x p l a i n e d  o r i g i n a l . ' By r e l a t i n g  and  o f each  specific  had a d i f f e r e n t  monomer  need  o f the range  can best  o f 50,000  affected  i n this  volume  o f t h e K_^_. t e r m  partial  concept  protein  this  When t h a t  the  between  f o r , a n d t h e u s e o f a mean  volume  to a portion  calculation.  system,  utilized  specific  However,  specific  assigned  This  approach  n o t be accounted  partial  up a  volume  developed by Scholte,  i n partial  required.  specific  I t must  procedure  under  Monroe  study,  be  1880  emphasized,  requires  some  so that the  149.  p a r t i a l s p e c i f i c volumes c o u l d be assigned  t o the c o r r e c t  m o l e c u l a r weight r e g i o n . The above procedure was not i n c l u d e d  i n the FORTRAN  program s i n c e the i n i t i a l o b j e c t i v e was t o compare the r e s u l t s with Scholte's  l i n e a r programming program t h a t  used lambda as a p r e c a l c u l a t e d use  i n p u t term.  In order t o  the above approach, lambda had t o be c a l c u l a t e d as  p a r t of the program.  150.  CONCLUSION  The i n i t i a l o b j e c t i v e of automating the u l t r a c e n t r i f u g e f o r the purpose o f o b t a i n i n g m o l e c u l a r weights for  both homogeneous and heterogeneous systems  technically successful. of  was  With t h i s system, r o u t i n e a n a l y s i s  p r o t e i n s c o u l d be made w i t h ease and r e l a t i v e l y  tedium.  little  However, the problem of c o r r e c t b a s e l i n e  d e t e r m i n a t i o n when u s i n g UV o p t i c s s t i l l to make the system more u s e f u l .  has t o be e l i m i n a t e d  Other workers  (7,46) have  r e c o g n i z e d the s i g n i f i c a n c e o f the b a s e l i n e problems, and a s o l u t i o n f o r d e t e r m i n i n g the t r u e b a s e l i n e was by Chernyak  and Magretova  (6).  This correction  suggested was  i n v e s t i g a t e d and shown t o be u s e f u l f o r homogeneous systems (.34), but not a p p l i c a b l e to heterogeneous systems. the  Until  b a s e l i n e problem i s overcome, the accuracy o f m o l e c u l a r  weight d e t e r m i n a t i o n s u s i n g the UV scanner i s open t o question. The use o f desktop computing f a c i l i t i e s allowed the a n a l y s i s o f the u l t r a c e n t r i f u g a l data t o be performed q u i c k l y and a c c u r a t e l y , u s i n g r e g r e s s i o n methods t h a t were out ago.  o f r e a c h of many s m a l l e r l a b o r a t o r i e s o n l y a few years These curve f i t t i n g  becoming  techniques are now  rapidly  the standard method of a n a l y s i n g a l l types o f  151.  experimental data desktop  calculator  mathematical functions, use.  The  desktop of  The  programmable  features  knowledge t o program complex  m a k i n g them r e a d i l y combination  calculator  available  of the a c q u i s i t i o n  facilitated  second  objective  the study of p r o t e i n s  comparison  mathematical for repetitive system  the s u c c e s s f u l  use  s t u d y was  Although for positive  the  desired  the approach  problems  prevented  t h e MWD  of  time.  calculation  somewhat e m p i r i c a l ,  that  any m o l e c u l a r  conclusion.  The  as u s e f u l  successfully  statistical as t h o u g h t  b u t came t o t h e  parameters  by Magar  were n o t  (31), but  implemented as a v e h i c l e  The  not  l i n e a r p r o g r a m m i n g method  such assumptions,  f o r an  the  weights  w i t h t h e n e g a t i v e f r e q u e n c y v a l u e s were The  in  regression  period  to s o l v i n g  f r e q u e n c y v a l u e s was  S c h o l t e made no  solution.  calculations  a distribution.  f o r an e x t e n d e d  of the s o l u t i o n .  t o be  processing  of multiple  l a u n c h e d when d r i v e  bore out the assumption  associated part  and  became o f m i n o r i m p o r t a n c e  method o f o b t a i n i n g  of the c e n t r i f u g e  results  o f u s i n g MWD  with the i n v e s t i g a t i o n  a s an a l t e r n a t i v e This  the  results. The  for  on  a l l o w a n y o n e w i t h a minimum o f  experimental u l t r a c e n t r i f u g a l data into  final  was  (16).  of  same found  were iterative  m a j o r d r a w b a c k t o t h e method o f S c h o l t e  t h e numerous s p e e d s  n e c e s s a r y f o r t h e MWD  to  be  152.  r e s o l v e d completely.  T h i s was  a genuine problem  since  a seven speed run, t h a t r e q u i r e d e q u i l i b r i u m to be reached at  each speed, was extremely time consuming and demanding  on the instrument.  Furthermore, the accuracy of the  d a t a was d i l u t e d c o n s i d e r a b l y by the repeated measurements t h a t r e q u i r e d r e d e f i n i n g of the meniscus and c e l l at  each speed.  bottom  The use o f the i t e r a t i v e procedure  t h e o r e t i c a l l y allowed the f r e q u e n c i e s o f the i n d i v i d u a l components t o be e l u c i d a t e d f r r m the d a t a o f a s i n g l e speed run.  One assumption was n e c e s s a r i l y made i n the  use of the i t e r a t i v e procedure:  the components were  s u f f i c i e n t l y separated t o form d i s t i n c t peaks. was  not p o s s i b l e , e s t i m a t e s of the m o l e c u l a r weights  s h o u l d be known. was  I f that  The r e s o l u t i o n o f the i t e r a t i v e method  shown t o be much g r e a t e r than t h a t of the complete  molecular w e i g h t . d i s t r i b u t i o n i t s e l f .  In g e n e r a l , the  d i s t r i b u t i o n c a l c u l a t i o n r e q u i r e d t h a t a f a c t o r o f two s e p a r a t e d the m o l e c u l a r weight o f the components. However, a s i g n i f i c a n t l y s m a l l e r f a c t o r was for  the i t e r a t i o n i f an e s t i m a t e o f the component  m o l e c u l a r weights was in  allowable  available.  T h i s was  illustrated  the case o f the model i n t e r a c t i o n o f lysozyme and  ovalbumin.  Undefined d i s t r i b u t i o n s were g e n e r a l l y  o b t a i n e d from s i n g l e speed d a t a and a smoothing was  routine  developed t o c o n v e r t these d a t a i n t o a more s u i t a b l e  153.  form. and  T h i s o p e r a t i o n , however, was  completely e m p i r i c a l  the t r u e d i s t r i b u t i o n would not n e c e s s a r i l y  be  represented. The  actual  calculation  i n t h i s work.  the p o t e n t i a l performed  using  an  of the complete  to r e a l p r o t e i n mixtures  investigated  be  application  accuracy  was  In o r d e r  not  evaluate  o f t h e method, e x p e r i m e n t s  o f known c a p a b i l i t y  i n terms o f o v e r a l l  accuracy  and  accuracy.  an  unknown  reliability,  r e a l p r o b l e m s were p o s e d  i n using this  instrument  assess  sophisticated  technique.  in  such  a relatively  the area of u l t r a c e n t r i f u g a t i o n  ultracentrifuge  t o be  a n a l y t i c a l work.  defeating  effort.  seemed t o p r o d u c e t h e obtain  a distribution,  were b a s i c a l l y  for  In r e s p e c t t o t h e equipment c a l c u l a t i o n became a  However, t h e  to Workers  c o n s i d e r the Model  t h e minimum s t a n d a r d  t h e e v a l u a t i o n o f t h e MWD  should  mixtures,  and  S i n c e t h e p r e p a r a t i v e u l t r a c e n t r i f u g e was quantity  fully  to t r u l y  on w e l l c h a r a c t e r i z e d p r o t e i n  instrument  MWD  few  E  any available, self  experiments  performed  fundamental i n f o r m a t i o n r e q u i r e d to and  i n l i n e with  indicated  t h a t the  results  t h e known c o m p o s i t i o n  of  the  sample. As weights  m e n t i o n e d p r e v i o u s l y , an e s t i m a t e  of the  or the range o f m o l e c u l a r  present  weights  molecular i n the  154.  sample u n d e r i n v e s t i g a t i o n w o u l d be the  MWD.  T h e r e were a number o f  available protein  that could  systems.  individual itself,  mixture.  This  reasonable  f i t c o u l d be  estimate  of  the  s i n g l e polynomial could  complete  be  the  molecular  the  component  data  concentration  Other techniques  of the  obtained, an  and  MWD  suffice  to obtain  distribution t h a t c o u l d be  estimate  other  of  high molecular  the  Y p h a n t i s method  t h e low m o l e c u l a r  Y p h a n t i s and  Roark  curve  (69),  fitting  orthogonal If  a  weights  i n turn  along  providing  calculation. i n a l l cases  A and  a good f i t o f  the  (26). u s e f u l are (17)  the  approach  that allows  w e i g h t component t o  s e r v i c e a b l e method i s t h e  u s e d by  or  (11,-38) .  t o e q u i l i b r i u m method o f A r c h i b a l d estimate  of  t o use  obtained,  range f o r the  required  conventional  conditions, of  made, t h e m o l e c u l a r  might not  mixture  de-  raw  s o l u t i o n c o l u m n c o u l d be  the  the  i n d i v i d u a l components and  t o smooth t h e  for  allowed  s u c h as m u l t i p l e r e g r e s s i o n  polynomials  methods  to run  the  either  A n o t h e r u s e f u l a p p r o a c h was  techniques  several  with  rotor, using  under i d e n t i c a l  the  and  e s t a b l i s h these parameters  or Yphantis c e l l s .  weights of  obtaining  m o s t o b v i o u s method was  i n a multiplace  termination,  an  The  techniques  components s i m u l t a n e o u s l y  centrifuge  the  help  useful in  an be  (68)  that could  yield  weight  component.  One  'two  species  later  plot' analysis  a p p l i e d to  protein  155.  m i x t u r e s by  J e f f e r y and  Pont  (25).  The  t h e s e a n c i l l a r y methods c o u l d h e l p s y s t e m and  e s t a b l i s h whether the  calculation  and  the  simplex  use  of  some o f  to c h a r a c t e r i z e  r e s u l t s of the  i t e r a t i o n were  the  MWD  reasonably  correct. Many w o r k e r s f o r o v e r f o r t y to obtain The  accurate  c o m p u t e r has  p r o b l e m s , as the  obtain  i n the  that  a MWD  the  of  means t h a t  and  solutions.  are  an  and  approach.  the to  a p e r f e c t MWD,  since  number o f  however, o n l y  Wiff  data. attempt  From t h e  and  the  best  to f i n d  these  to  number Donnelly, limited  regressioni t has  the m u l t i p l e  been.  re-  multiple  advantage of  f i t frequencies  the  methods o f  The  the  solutions  p a r a m e t e r s a v a i l a b l e t o be  s o l v i n g f o r the  is  This  s t u d y o f model d a t a  Scholte  been  a limited  The  to  guarantee  itself  s o l u t i o n , however, has  statistical  does not  i s b e c a u s e i t has  a p p r o a c h work e q u a l l y w e l l .  regression  However,  Rinde equation  infinite  old  t h e methods u s e d  i s a l s o t r u e of the m u l t i p l e  shown t h a t t h e method o f gression  This  e x p l a i n the  Gehatia  solution to  Improperly Posed Problem.  In r e a l i t y ,  This  attempted  Rinde e q u a t i o n .  Rinde equation  that the  i s an  solutions w i l l  simplex  the  to obtaining  there  equation.  Scholte  methods o f  s o p h i s t i c a t i o n of  late  Rinde equation  of  case of  from the  major d e t e r r e n t  the  new  problem i s solved.  recognized  have  MWDs f r o m u l t r a c e n t r i f u g a l d a t a .  allowed  computational  years  and  u s e d as  allowing a  molecular  guide  156.  weights.  This  theoretically  advantage  allows the evaluation  w e i g h t s and c o n c e n t r a t i o n a t one r o t o r is  speed.  i f the multiple  o f the molecular  o f t h e components o f a m i x t u r e  Whether t h i s  unknown, b u t c e r t a i n l y  required  i s substantial, since i t  i s possible  fewer r o t o r regression  in reality  speeds w i l l  approach  be  i s used.  157.  LITERATURE  CITED  Adams, E.T. J r . , W.E. Ferguson, P.J. Wan, J.D. S a r q u i s , B.M. E s c o t . 1975. Some modern aspects o f u l t r a c e n t r i f u g a t i o n . Separ. S c i . 10:17 5v, Adams, E.T. J r . , P.J. Wan, D.A. Soucek, and G.H. Barlow. 1973. M o l e c u l a r weight d i s t r i b u t i o n s from s e d i m e n t a t i o n e q u i l i b r i u m experiments. Polymer M o l e c u l a r Weight Methods. Advan. Chem. Ser. 125:235. Bird,  C. U.B.C. Documentation. System/360 Scientific Subroutine Package, V e r s i o n I I I , Programers Manual, IBM Form No. 6H20-0205-4.  Bowen, T . J . 1970 An I n t r o d u c t i o n t o U l t r a c e n t r i f u g a t i o n . W i l e y - I n t e r s c i e n c e , London. Box,  G.E.P. and N.R. Draper. Evolutionary Operation.  1969. John W i l e y , New  York.  Chernyak, V. Ya. and N.N. Magretova. 1975. A new approach t o m o l e c u l a r weight c a l c u l a t i o n from sedimentation e q u i l i b r i u m experiments. Biochem. and Biophys. Res. Comrnun. 65:3. Chervenka, C.H. E v a l u a t i o n of a UV scanning a c c e s s o r y f o r preparative ultracentrifuges. Applications Data Beckman Instruments Spinco DS-452. Chervenka, C.H. 1973. A Manual of Methods f o r the A n a l y t i c a l C e n t r i f u g e Beckman Instruments, P a l o A l t o , C a l i f . C l a r k e , R. and S. Nakai. 1971. I n v e s t i g a t i o n of K - a ^ - c a s e i n i n t e r a c t i o n by fluorescence p o l a r i z a t i o n . B i o c h e m i s t r y 10:3353 s  C l a r k e , R. and S. Nakai. 1972. F l u o r e s c e n t s t u d i e s o f K - c a s e i n w i t h 3anilinonaphthalene-l-sulphonate. Biochim. e t Biophys. A c t a 257:61.  158. 11.  C r e e t h , J.M. and R.H. P a i n . 1967. The d e t e r m i n a t i o n o f m o l e c u l a r w e i g h t s o f b i o l o g i c a l m a c r o m o l e c u l e s by u l t r a c e n t r i f u g e m e t h o d s . Progr. i n B i o p h y s . and M o i . B i o l . 17:217.  12.  C r e p e a u , R.H., S . J . E d e l s t e i h  a n d M.J.'Rehmar.  1972 .  Analytical ultracentrifugation with absorption o p t i c s and a s c a n n e r - c o m p u t e r system. Anal. Biochem. 50:213. 13.  D a v i s , B . J . 1964. D i s k e l e c t r o p h o r e s i s . I I . Method and a p p l i c a t i o n t o human serum p r o t e i n s . Ann. N.Y. A c a d . S c i . 121:404.  14.  D o n n e l l y , T.H. 1966. The d i r e c t e s t i m a t i o n o f c o n t i n u o u s m o l e c u l a r w e i g h t d i s t r i b u t i o n by e q u i l i b r i u m u l t r a c e n t r i f u g a - r tion. J . P h y s . Chem. 70:1862.  15.  D o n n e l l y , T.H. 1969. Some c a p a b i l i t i e s a n d l i m i t a t i o n s o f t h e L a p l a c e t r a n s f o r m method f o r t h e d i r e c t e s t i m a t i o n o f continuous m o l e c u l a r weight d i s t r i b u t i o n s from equilibrium ultracentrifugation. A n n . N.Y. A c a d . Sci. 164 ( A r t . 1 ) : 1 4 7 .  16.  D r a p e r , N.R. a n d H. S m i t h . 1966. A p p l i e d Regression A n a l y s i s . John New Y o r k .  Wiley,  17.  E r l a n d e r , S.R. and J . E . F o s t e r . 1959. Application of the Archibald sedimentation p r i n c i p l e to p a u c i d i s p e r s e macromolecular systems. J . of Polymer S c i . V o l . XXXVII:103.  18.  F u j i t a , H. 1975. Foundations of U l t r a - C e n t r i f u g a l A n a l y s i s . Wiley, Toronto.  John  19.  F u j i t a , H. 1962. Mathematical Theory o f Sedimentation A n a l y s i s . A c a d e m i c P r e s s , New Y o r k .  20.  F u j i t a , H. and J.W. W i l l i a m s . 1966. S e n s i t i v i t y of sedimentation equilibrium data t o solute polydispersity. J . P h y s . Chem. 70:309.  21.  G e h a t i a , M. and D.R. W i f f . 1970. Solution of F u j i t a ' s equation f o r e q u i l i b r i u m s e d i m e n t a t i o n by a p p l y i n g T i k h o n o v ' s regularizing functions. J . o f Polymer S c i . P a r t A-2. 8:2039.  159. G e h a t i a , M. and D.R. W i f f . 1972. Determination o f a m o l e c u l a r weight d i s t r i b u t i o n from e q u i l i b r i u m sedimentation by a p p l y i n g r e g u l a r i z i n g f u n c t i o n s . Experimental v e r i f i c a t i o n . European Polymer J . 8_:585. Gehatia, M. and D.R. W i f f . 1973. A new way o f d e t e r m i n i n g m o l e c u l a r weight d i s t r i b u t i o n i n c l u d i n g low m o l e c u l a r weight from e q u i l i b r i u m sedimentation. Polymer M o l e c u l a r Weight Methods. Advan. Chem. S e r . 125. Gibbons, R. A., S.N. Dixon and D.H. Pocock. 1973. A new g e n e r a l method f o r t h e assessment o f t h e molecular-weight d i s t r i b u t i o n o f p o l y d i s p e r s e preparations. I t s a p p l i c a t i o n t o an i n t e s t i n a l e p i t h e l i a l g l y c o p r o t e i n and two d e x t r a n samples, and comparison w i t h a monodisperse g l y c o p r o t e i n . Biochem. J . 135:649. J e f f r e y , P.D. and M.J. Pont. 1969. The e s t i m a t i o n o f m o l e c u l a r weights i n mixtures of two p r o t e i n s by t h e meniscus d e p l e t i o n method. Biochemistry 8_:4597. Kar, E.G. and K.C. Aune. 1974. Analyses o f sedimentation e q u i l i b r i u m d a t a . A n a l . Biochem. 62:1. Kason, W.R. and S. Nakai. 1971. Interaction of a .-casein with polyethylenimine. J . Dairy S c i . 54:461. S  K o n i n g s v e l d , R. and C.A.F. Tuijnman. 1959. C a l c u l a t i o n of the m o l e c u l a r weight d i s t r i b u t i o n of polymers from experimental d a t a . J . o f Polymer Sci. XXXIX:445 L a n s i n g , W.D. and E.O. Kraemer. 1935. M o l e c u l a r weight a n a l y s i s of mixtures by sedimentation e q u i l i b r i u m i n the Svedberg ultracentrifuge. J . Am. Chem. Soc. 57:1369. Magar, M.E. 1970. M o l e c u l a r weight d i s t r i b u t i o n f u n c t i o n s and v i r i a l c o e f f i c i e n t s from sedimentation e q u i l i b r i u m d a t a . J . Theor. B i o l . 27:127. Magar, M.E. 1972. Data A n a l y s i s i n B i o c h e m i s t r y and B i o p h y s i c s . Academic P r e s s , London.  160. 32.  Moore, J.B. 1975. F o r t r a n Programming w i t h the WATFIV Compiler. Reston P u b l . Co. Inc., Reston, V i r g i n i a .  33.  Morgan, S.L. and S.N. Deming. 1974. Simplex o p t i m i z a t i o n of a n a l y t i c a l methods. A n a l . Chem. 46:1170.  34.  Nakai, S. Unpublished  35.  36.  chemical  results.  Nakai, S. and C M . Kason. 1973. D e t e r m i n a t i o n of a s s o c i a t i o n c o n s t a n t s of K-a  ,s1  casein interaction. Presented a t the 68th Annual Meeting, Amer. D a i r y S c i . Assoc. Nakai, S. and C M . Kason. 1974. A f l u o r e s c e n c e study of the i n t e r a c t i o n s between • K- and a ^ c a s e i n and between lysozyme and ovalbumin. Biochim. et Biophys. A c t a 351:21.  37.  N e l d e r , J.A. and R. Mead. 1965. A simplex method f o r f u n c t i o n m i n i m i z a t i o n . Computer J . 2:308.  38.  N i e l s e n , H.C and A.C. Beckwith. 1971 P r o t e i n aggregation as s t u d i e d by sedimentation equilibrium. Recent developments i n instrumentat i o n and theory. J . Agr. Food Chem. 19:665.  39.  Osterhoudt, H.W. and J.W. W i l l i a m s . 1965. Sedimentation e q u i l i b r i a i n p o l y d i s p e r s e pseudoi d e a l s o l u t i o n s and a t low c e n t r i f u g a l f i e l d s . J . Phys. Chem. 69:1050.  40.  Payens, T.A.J. •Personal communication . Netherlands I n s t i t u t e f o r D a i r y Research (NIZO), Ede, The N e t h e r l a n d s . 1  41.  Provencher, S.W. 1967. Numerical s o l u t i o n of l i n e a r i n t e g r a l equations of the f i r s t k i n d . C a l c u l a t i o n of m o l e c u l a r weight d i s t r i b u t i o n s from s e d i m e n t a t i o n e q u i l i b r i u m d a t a . J . Chem. Phys. 46:3229.  42.  Provencher, S.W. and W. Gobush. 1968. Two methods f o r the c a l c u l a t i o n of m o l e c u l a r weight d i s t r i b u t i o n s from s e d i m e n t a t i o n e q u i l i b r i u m d a t a . Nat. Res. C o u n c i l P u b l . (US) #1573:143.  161. 43.  R e i n h a r d t , W.P. and P.G. S q u i r e . 1965. Computer a n a l y s i s of s e d i m e n t a t i o n e q u i l i b r i u m data from p a u c i d i s p e r s e and i n t e r a c t i n g systems. Biochim. Biophys. A c t a 94:566.  44.  Rinde, H. 1928. Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of Uppsala.  45.  Schachman, H.K. 1959. Ultracentrifugation i n Biochemistry. P r e s s , New York.  46.  Schachman, H.K. and S.J. E d e l s t e i n . 1973. U l t r a c e n t r i f u g a l studies with absorption optics and a s p l i t beam p h o t o e l e c t r i c scanner, i n Methods i n Enzymology, XXVII:3. E d i t e d by Ch. W. H i r s and S.N. Timasheff. Academic P r e s s , New York.  47.  S c h o l t e , Th. G. 1968. M o l e c u l a r weights and m o l e c u l a r weight d i s t r i b u t i o n of polymers by e q u i l i b r i u m u l t r a c e n t r i f u g a t i o n . P a r t I I . M o l e c u l a r weight d i s t r i b u t i o n . J . of Polymer S c i . , P a r t A-2 6_:111.  48.  S c h o l t e , Th. G. 1970. ,Determination of the m o l e c u l a r weight d i s t r i b u t i o n of polymers from e q u i l i b r i a i n the u l t r a c e n t r i f u g e . European Polymer J . 6_:51.  49.  Sober, H.A. 1968. Handbook o f B i o c h e m i s t r y . S e l e c t e d Data f o r M o l e c u l a r B i o l o g y . The Chemical Rubber Co., C l e v e l a n d , Ohio.  50.  Soucek, D.A. and E.T. Adams J r . 1976. M o l e c u l a r weight d i s t r i b u t i o n s from s e d i m e n t a t i o n e q u i l i b r i u m of n o n i d e a l s o l u t i o n s . J . of C o l l o i d and I n t e r f a c e S c i . 55:571.  51.  Spendlay, W., G.R. Hext and F.R. Himsworth. 1962. S e q u e n t i a l a p p l i c a t i o n of simplex d e s i g n s i n o p t i m i z a t i o n and e v o l u t i o n a r y operation.. Technometrics 4_:441.  52.  Sundelof, L.A. 1968 Determination of m o l e c u l a r weight d i s t r i b u t i o n s from s e d i m e n t a t i o n - d i f f u s i o n e q u i l i b r i u m experiments by a c o n v o l u t i o n procedure. Ark. Kemi. 23:279.  53.  Svedberg, T. and J.B. N i c h o l s . 1926. The m o l e c u l a r weight of egg albumin. I . In e l e c t r o l y t e - f r e e c o n d i t i o n . J . Am. Chem. Soc. 48:3090.  Academic  162.  Van  H o l d e , K.E. a n d R.L. B a l d w i n . 19 5 8 . Rapid attainment o f sedimentation e q u i l i b r i u m . J . P h y s . Chem. 6 2 : 7 3 4 .  W a l e s , M. 1948. Sedimentation e q u i l i b r i a o f p o l y d i s p e r s e nonideal solutes. I . Theory. J . Phys. C o l l o i d Chem. 5 2 : 2 3 5 . W a l e s , M. 1951. Sedimentation e q u i l i b r i a o f p o l y d i s p e r s e nonideal solutes. Uses and l i m i t a t i o n s o f t h e equilibrium ultracentrifuge. J . Phys. C o l l o i d Chem. 5 5 : 2 8 2 . W a l e s , M., F . T . A d l e r a n d K . E . V a n H o l d e . 1951. Sedimentation e q u i l i b r i a o f p o l y d i s p e r s e nonideal solutes. V I . Number-average m o l e c u l a r weight and m o l e c u l a r - w e i g h t d i s t r i b u t i o n f u n c t i o n s J . P h y s . C o l l o i d Chem. 5 5 : 1 4 5 . W a l e s , M., J.W. W i l l i a m s , J . O . T h o m p s o n a n d R.H. Ewart. 1948. Sedimentation e q u i l i b r i a o f p o l y d i s p e r s e nonideal solutes. J . P h y s . Chem. 5 2 : 9 8 3 . Wan,  P . J . 1974. M o l e c u l a r weight d i s t r i b u t i o n s by u l t r a c e n t r i f u g a t i o n 1 9 7 3 . F r o m D i s s . A b s t . I n t . B. 3_4 (12) : 5 8 4 6 .  W i f f , D.R. a n d M. G e h a t i a . 1971. T e c h n i c a l R e p o r t , AFML-TR-67-121, P a r t Airforce Materials Laboratory. Wright A i r Force Base, Ohio.  V. Paterson  W i f f , D.R. a n d M. G e h a t i a . 1972. Determination o f molecular weight distribution by a p p l y i n g a m o d i f i e d r e g u l a r i z a t i o n t e c h n i q u e . J. Macromol. Sci.-Phys. B6(2):287. W i f f , D.R. a n d M. G e h a t i a . 1973. Techniques used i n a p p l y i n g r e g u l a r i z a t i o n t o the i l l - p o s e d problem o f determining a m o l e c u l a r weight d i s t r i b u t i o n from s e d i m e n t a t i o n e q u i l i b r i u m J . P o l y m e r S c i . Symposium No. 4 3 : 2 1 9 . W i f f , D.R. a n d M. G e h a t i a . 1976. I n f e r r i n g a molecular weight d i s t r i b u t i o n , an i l l - p o s e d problem; and e s t a b l i s h i n g t h e m o l e c u l a r weight scale using magnetic float techniques. B i o p h y s i c a l Chem. 5_:199.  163. W i l l i a m s , J.W. 1972. U l t r a c e n t r i f u g a t i o n o f Macromolecules; Topics. A c a d e m i c P r e s s , New Y o r k .  Modern  W i l l i a m s , J.W. a n d W.M. Saunders. 1954. Size d i s t r i b u t i o n a n a l y s i s i n plasma extender systems. I I . Dextran. J . P h y s . Chem. 58:854. W i l l i a m s , J.W. a n d W.M. Saunders. 1954. Size d i s t r i b u t i o n a n a l y s i s i n plasma extender systems. I. Gelatin. J . P h y s . Chem. 58:774. W i l l i a m s , J.W. a n d K . E . V a n H o l d e . 1958. The t h e o r y o f s e d i m e n t a t i o n a n a l y s i s . Rev. 38:715. Y p h a n t i s , D.A. 1964. Equilibrium ultracentrifugation Biochemistry 3_:297.  Chem.  of dilute  solutions.  Y p h a n t i s , D.A. a n d D . E . R o a r k . 1971. E q u i l i b r i u m c e n t r i f u g a t i o n o f non-ideal systems. The Donnan e f f e c t i n s e l f - a s s o c i a t i n g s y s t e m s . Biochemistry 10:3241.  164.  APPENDIX  A.  P r o t e i n s Used i n t h i s  Investigation  Manufacturer  Protein  B.  Bovine serum albumin  Sigma  Catalase  Sigma  Conalbumin  Sigma  Lysozyme  Worthington  Ovalbumin  ICN  Thyroglobulin  Sigma  Trypsin  Calbiochem  Inhibitor  Pharmaceuticals  Assembly o f the Yphantis C e l l s  The  Yphantis c e n t r i f u g e c e l l  required  special  handling,  s i n c e i t d i d n o t have any f i l l i n g h o l e s and tended t o l e a k i f the f o l l o w i n g procedure was n o t f o l l o w e d . the Yphantis c e l l ,  When assembling  two window gaskets and two screw r i n g  gaskets should be used r a t h e r than one.  The c e l l  was  assembled empty, the screw r i n g torqued t o 110 l b . - i n . ( 9 1 6 newton-centimeters), and the f i l l i n g h o l e p l a c e a g a i n s t the c e n t e r p i e c e w a l l .  screws p u t i n t o  The screw r i n g was  then r e l e a s e d and the top window and h o l d e r removed by l i g h t tapping  against a s o l i d surface.  c o u l d then be f i l l e d  with a syringe.  The c e l l  sectors  A f t e r f i l l i n g , the  165.  window and  h o l d e r were c a r e f u l l y  reinserted  screw r i n g  torqued  The  were t h e n  to 110  r e l e a s e d , the  newton-centimeters) If  t h i s procedure  and  was  lb.-in.  screw r i n g the  C.  P r o g r a m s f o r t h e Monroe  incidence of  description  1.  estimate  t where:  H o l d e and  of the time  equilibrium.  The (b - m)  e  U(a) a  =  screws  t o 135  l b . - i n . ( 1125  screws r e t i g h t e n e d . c o u l d be  run  at  leakage.  Calculator  be  p r o v i d e d below, a l o n g  equilibrium  T h i s p r o g r a m was  an  hole  in with  use.  Time t o r e a c h  d e r i v e d by Van  the  o f t h r e e programs mentioned  t h e t e x t o f t h i s work w i l l of t h e i r  hole  followed, the c e l l  speeds w i t h o u t  a note  retorqued  filling  high  A brief  filling  and  =  program  w r i t t e n to solve the  Baldwin  (54)  relation  i n order to  r e q u i r e d f o r a system to  provide reach  relation i s : 2  F(a)/D 1 +  (l/4^ a ) 2  2  R T / M ( 1 - v p ) r (b -  m)  Eq.  IA  Eq.  2A  Eq.  3A  Eq.  4A  Figure  Input:  IA.  Time t o r e a c h e q u i l i b r i u m  A - diffusion  coefficient  -e - molecular weight  - 1 - vp -  rpm  -  r  Output: B - time i n hours  program  167  5 « JO.0  0 0 0 0 0 0  1 <. 0 0 0 0 0 0 0 0 0 a -• 6 1 8 0 0 0 0 0 0 2 «. 6 8 0 0 0 0 0 0 0 1 <. 2 0 0 0 0 0 0 0 0 7 .000000000  4  . 0 0 0 0 3  2  •43734  0  0 0 0  5 8 0 0  4-059700503  1 -0 3 0 4 -0  -0 1  C4 0 0 -0  }  1 0  0 1  A  168.  and  2  D d i f f u s i o n c o e f f x c i e n t (F.icks - cm b - c e l l bottom  -1  -7  sec  x 10  )  m - meniscus r - (b +  m)/2  e - measurement of the departure  from e q u i l i b r i u m  By knowing the l e n g t h of the s o l u t i o n column and estimated  molecular  weight and d i f f u s i o n  the  coefficient,  the time t o reach e q u i l i b r i u m c o u l d be determined.  The  e r r o r term £ r e f e r s to the d e v i a t i o n of the p a t t e r n from i t s t r u e e q u i l i b r i u m p o s i t i o n and was 0.001.  The  in Figure  2.  i n p u t and output of t h i s program are  presented  IA.  Orthogonal polynomial T h i s was  T h i s curve f i t t i n g  r e q u i r e d t h a t the data be o r i g i n a l program was  one  data c o u l d be obtained  provided modified  program  technique  spaced e q u a l l y along  c a l c u l a t o r company, but was  the c a l c u l a t i o n .  curve f i t t i n g program  an a l t e r n a t i v e curve f i t t i n g  to m u l t i p l e r e g r e s s i o n .  The  u s u a l l y set at  the  x-axis.  by the Monroe so t h a t the b e s t f i t  from the c o e f f i c i e n t s d e r i v e d  from  In g e n e r a l , t h i s program worked very  w e l l but l a c k e d the automated c a p a b i l i t y and parameters provided  statistical  by the m u l t i p l e r e g r e s s i o n program.  169.  3.  Data conversion This  program f o r m a n u a l l y  p r o g r a m was w r i t t e n  comparison o f data manually scan t o t h a t The  obtained  from t h e e q u i l i b r i u m  program produced absorbance v s . r a d i a l radial  simultaneously,  and s t o r e d  distance  groups o f r e g i s t e r s . compared  input  D.  selected  distance,  i n three  Any s e t o f d a t a c o u l d ,  therefore,  t o t h e a c q u i s i t i o n d a t a by p l o t t i n g .  and o u t p u t o f t h i s p r o g r a m a r e p r e s e n t e d  FORTRAN  multiple  optimization  routine.  incorporated  into  Pertinent in  t h e MWD  and t h e s i m p l e x  p r o g r a m , i t was  optimization  was  originally  i tcould  be r e a d i l y  problems.  f o r r u n n i n g each program w i l l  3A i l l u s t r a t e s  t h e MWD  t h e deck o r d e r  calculation information  reference  2A.  the complete  In the  s e c t i o n t h e d e c k a r r a n g e m e n t and p e r t i n e n t  instructions  of  i n Figure  Although the simplex r o u t i n e  f o r use i n other  following  Figure  regression,  as a s e p a r a t e program, so t h a t  modified  The  Programs  program u s i n g  written  data  separate  Two m a j o r FORTRAN p r o g r a m s were w r i t t e n ; MWD  system.  squared and'c(£)/c  the data  data  direct  from t h e d a t a a c q u i s i t i o n  absorbance v s .  be  to allow  selected  to this  be p r e s e n t e d .  f o r the three  and f o r t h e s i m p l e x  routine.  about the deck i s e x p l a i n e d figure.  modes  below,  Input and output of the data c o n v e r s i o n program f o r manually s e l e c t e d d a t a .  Figure  2A.  Input:  A - number of data  points  - number o f c h a r t v a l u e s between r e f e r e n c e edges - cell  bottom  - meniscus - initial  concentration  B - c h a r t u n i t s from the meniscus t o t h e c e l l bottom C - s e l e c t e d absorbance v a l u e s Output: D - r a d i a l d i s t a n c e i n cm  2 E - x - In A F - ? - c(£)/c  171.  1 2 .0 0 0 0 8 5 • cooo 7 .9 • 4 0 0 0 • 6 00 0 6 3 < 7 6 -- 4 0  00 00  • 00 2 8 5 « .7 «• 1 9 4 5 6 «> 8 9 7 1 15-> 9 0 0  0  7 «• 1 3 8 1— 5 0 0« 0 0 0 0 -  _l  B  C D 2 0 0  0 0 0 0  5 0  9 5 2 7  6  2 14 6  7« 119 4 4 0 3 -• 0 0 0 0  5 0  6 8 6 0  5  9 9 8 9  7 -• 0 8 2 0 2 9 1« 0 0 0 0  5 0  4 2 0 0  5  8 0 8 1  7 • 0 6 3 2 2 3 9 -•  C0 0 0  !•: 1 10 • 0 0 0 0  7 «- 1 0 0 7 3 3 3 «• 0 0 0 0  : —I  5 0  15 4 7  5  6 7 3 3  4 9  8 9 0 1  5  4 7 6 4  0  19 3 1  1  7 5 4 3  0  2 5 6 7  1  4 14 0  0  3 2 0 2  1  16 8 4  0  38 3 5  1  0 2 10  0  4 4 6 6  0  8 3 8 5  yxyxyxxxyxxyxyyxyyyyyxyxyyyxxxxxxxxxxxxxxxxxxxyxxxxxyxyxxxxxxxxxxxxxxxxxxxxxxxxx C  JOB  SETUP  FOR THE V A R I O U S  MODES  O F T H E MHO  CALCULATION I l l u s t r a t i o n o f card deck f o r v a r i o u s modes o f t h e MWD p r o g r a m a n d f o r t h e s i m p l e x program.  NOR M A i — c - * t r C t i t r * - T - t ' ) - ^ ' O O E N  tSTUNON TO"" PASSWRD -juiit "nJ.S3unp»K*onj.S!MuO? u  -I5ono 0,o03nF-na O.JSSS H f S ' OF OA! A 0,003n F - n u — o ; n o o n  |6»-KIJMAT  u  ?\  ?.. n  jtso  -xono  2,SS;>  1  - n-, n o n  CARD t -CAPO-8CARD 3  i)  v'.n  -t  ~  CARO  "  a  »EMlF u. l »SIGN(IFK  ~c  —  ITERATIVE  J<!TGNP»« SPUN  TOMM  —- —  HOOF  --  noJ.PSUBPAK^OHJ.SIM^f)? nono  O.ootnF-ni 0.?5S3 RF S T OF T A T A o.wo <iiF-nu o.nnno no 2 a  (on-KlJMAT  —evn-  r.o  7.?00  .2«373uSflEtl  i 0 , n0n 1 ooon  "CARD"1 C*RU 2 CARD 3 CARD CARO  3nooo  "TENDFTLE »3TGkJHFF  c -SMOOTHING-MODE  r  -  1STSM0N TO*» PASSXORO— «RIIN o q J . S S l i 8 P » K * 0 B J . S I w O ? 1 6 3 - K I J M A T u  u  u  ? I  isnno ,?S (1.0030E-011 0.?5SJ RES' O F PA T A 0,oit«nF«nu O.ooooF-na 0 ,»030E-ou  I  t « JE^OFILE  2.0  -Anno ?,S5?  CARD 1 CARD 2 CARD 3  ?  ~ ? 5 o t ' . f l 1  6.Ron  7,700  .?aj7Sfl58EII  T  •TTTnTVlVTfo,nooo o , nono 0,nooo n i lo  0.?5n n.?sn n.oon" n o  A P CARD CARD CARD CARD  O T 5b 7 8  CARD  I  •JSTGVOFF  C SiMBLFx  PROGRAM  $3TGW0N T P M P»SS*OBO ' J R U ^ np,j;si"Pir"* 8 It2 32  3  JEW" FILE JSTGHOFF  c  6  mono  2n000  ^1  70000  7UU  anooo  ?nonoo  500000  C A R D ' ?.  173.  1.  The m o l e c u l a r weight d i s t r i b u t i o n program  a.  C a l c u l a t i o n mode  Card 1 c o n t a i n s the f o l l o w i n g the  number o f m u l t i p l e s  information: of the s t a r t i n g  m o l e c u l a r weight f o r t h e s e r i e s the  number o f data cards  the  number o f c y c l e s p  -  the m o l e c u l a r weight i n t e r v a l  -  the c a l c u l a t i o n mode i n d i c a t o r ; where 0 = MWD  c a l c u l a t i o n mode,  1 = i t e r a t i v e mode and 2 = smoothing mode  Card 2: the  s t a r t i n g m o l e c u l a r weight o f a s e r i e s  the  power v a l u e o f t h e m o l e c u l a r weight  multiple the  rotor  speed  the p a r t i a l s p e c i f i c volume the  density  the  cell  meniscus  the  cell  bottom  RT  o f the s o l v e n t  174.  Only the f i r s t to  two terms were r e q u i r e d f o r the c a l c u l a t i o n  proceed, w i t h the r e s t b e i n g f o r r e f e r e n c e o n l y .  Card 3: -  X  -  K c(U/c_ o p t i c s a l t e r n a t i v e , where 1 = UV o p t i c s , 2 = Schlieren  Card 4: -  A s p e c i a l n u l l s e t r e q u i r e d a t t h e end of  each data s e t t o f o r c e the i n t e r c e p t  towards zero.  b.  I t e r a t i v e mode  The deck arrangement was s i m i l a r t o the c a l c u l a t i o n mode except t h a t i t c o n t a i n e d an a d d i t i o n a l card.  Card 1 changed i n meaning somewhat, w i t h the f i r s t  term now r e f e r r i n g t o the number o f components c o m p r i s i n g the d i s t r i b u t i o n ,  and t h e t h i r d term becoming a c o n s t a n t  t h a t has t o s t a y a t a v a l u e o f one. Card 2 was a c t u a l l y redundant,  b u t was kept i n the deck.  Card 5 was the  i t e r a t i o n c o n t r o l card f o r t h e simplex o p t i m i z a t i o n r o u t i n e ,  175.  where t h e f i r s t term d e f i n e d the maximum number o f i t e r a t i o n s , t h e second  the number o f components i n t h e  system, t h e t h i r d t h e number o f m o l e c u l a r  weights  a s s o c i a t e d w i t h the components and t h e r e s t t h e range l i m i t s a s s o c i a t e d with t h e components.  c.  Smoothing mode  Again the deck arrangement was s i m i l a r t o the c a l c u l a t i o n mode, except i t c o n t a i n e d e x t r a dummy cards t h a t served t o strengthen the frequency response from the i t e r a t i v e c a l c u l a t i o n .  obtained  Cards 4-6 were the dummy  cards t h a t contained t h e frequency v a l u e s i n the p o s i t i o n found t o be optimum by the i t e r a t i o n .  Card 8 was t h e  m a t r i x c a r d , w i t h t h e f i r s t term d e f i n i n g t h e number o f components i n t h e system, t h e second  t h e number o f terms  i n t h e matrix and t h e r e s t the a c t u a l m a t r i x v a l u e s which had t o be composed o f zeroes and ones.  This particular  deck was s e t up f o r a three component system, t h e optimum f r e q u e n c i e s o f which were found t o be 0.50, 0.25 and 0.25. The  a c t u a l t e r m i n a t i n g matrix would have appeared a s :  1  1  0  0  0  0  0  0  1  1  0  0  0  0  0  0  1  1  176.  But was converted  to a linear vector:  100/100/010/010/0 01/001  A two component system was handled i n a s i m i l a r  2.  manner.  The simplex  program  The  a l g o r i t h m was o r i g i n a l l y w r i t t e n as  simplex  a separate program, designed  t o simulate  the a c t i o n o f  going through the m u l t i p l e r e g r e s s i o n by p i c k i n g up a v e c t o r o f dummy F - r a t i o v a l u e s t h a t were read i n .  Thus,  the program c o u l d be m o d i f i e d by removing t h e read statement and having  i t c y c l e through whatever o p e r a t i o n o r  c a l c u l a t i o n which produces the response term t o the f a c t o r s . Card 1 d e f i n e d t h e number o f dummy response v a l u e s w i t h t h e dummy v a l u e s .  along  Card 2 d e f i n e d t h e number o f  f a c t o r s , the number of range l i m i t terms and t h e range limit pairs.  A f t e r o b t a i n i n g t h i s i n f o r m a t i o n , t h e program  c o u l d f o l l o w through the a l g o r i t h m p i c k i n g up dummy response v a l u e s decisions.  and making t h e a p p r o p r i a t e  simplex  Listing  of  the  MWD  program  178.  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UJ  ca  IM  _  • J O  JJ X  ar _  o at  zz  >zz < _ o : or _ <  ac o  V)  z o  (X  ra — _ Q z uil < I  < to  1  1  u u u d v i  u  J  u  u  o  u  o  *  182.  oLcU  RO  _H O  4 — II w- jj> I— W Q U I I-*  _  O  z <  Ul  • i  1  >  o in m tn m i r i m  19  Ul  _  z  i  o o co co a  • __.-^  Q a  !  ! U. Z L v | • — UJ o "| - ru [ C uid s • u a i i ' o o H Q O I a o *I_LL or >--J_ o in rr o  •• o o  u . u. •ru x ». ru • •x I- in  »_ 3 5  I • im— i LU w a UJ < « oz] o m — z _ II _ o CT| _ Z  _ -t  ui  nH  3 O  _  cc z  T O O il S _ 3  3  O T J  u._ o  u. _  _ Hu .z  as •  U . QC —V3( OU. •  u  3 i •ui:  c  0  -m 3 T C l i n *— U i ~  l / l Z; 4. < o n I — _ _ H I Z| lac cr H O 3 O P O l L S  in — -  1  c o  o«r  It  «  •  - on  S Ujj V S .  z  ax|_ • q a o m c| o 3  rr  u..rJ o — H l i i ouji- • » o . - a ' - r> •*>. -  r,  n  ,—  _  •U  i  _  N.  I. N. N.  V.  *  N> •  rr < _ i 3  i I — —  _ r>  3 1  >-  o  _ _ 3  CD  _ _  m in Ul a a o <  ul > z o o —  Ul >  _lto <  u  _  ui a _i iiV) 4  or  i  _j • o (O  o t~ u  Ul  Ul  4 0 I  2 3 0 T Ul  >  o uiz z  J  oco  u m-ia ^> M — Z UJ c i _ « O 3 ll « - > z  • _ < X • < X »- _ i . — _ J - Z 3"£ — • •_ _ a 4 _'J < 1 r~ y i • _ u. ar. — v n or. o ar x oui z  • rr  — o  I-  I _  -> - J . •„  •• 1a _ u  _ z <  _ 2 3 Z  • _ | Q 4 O 3 Z> IT, U ) 4 ; • I Ij- - _ O U . • • s m in c - j r- • • m - • o m i — o in in -< - t / i . i a. in H xi H — in otninu.  w  _ _ z o  i  ui  _ _ i  ru < >- d — — z > a_ u a o _ uo c - . o  • •  z  • » UJI «  • >.  O  >  a  5  —jar Ul > z 3 d _ « a •31J z s _ < _ at 3 ->t-3CJ - _ CC n z LU-.3 _ I O _ _ _ O H * U . _ l  u  , 2  ru  I—ac  o z  a  o o  ut-z  _H • ^ — — <t» — + Z H + — n >4 -nJ _ » t-j — H <  it r 1 3 — H  x Ul  u 3  >-z  o Hn  z > _  _ > o  o  4 O a z O 3 o a - i a in —. u j r i H-  » -  -1 >  x O 4 T S Ul a in JJ3T. U ul 3 33  t  «(-  1 £ £0 -1 z a  n i-  _£ Z  a  iC  uuuc  u  i  U 1 3 arz  -1  VJ  30  L2*L2+K  —  c c c c c c  •  A D V A N C E S ' S T A R T I N G P O S I T I O N ANO E N D I N G P O S I T I O N OF K I J M A T NO* A H £ H E A D Y TO S T A R T ON A N O T H E R B L O C K OF K I J M A T ( N E X T R O T N )  3 J 3 3 1 2  43  I F ( K L I K . N E . l ) GO TO 3 1 2 *RITE<6,333> FORMA T ( • • 1 2 b ( I M * ) i / ) _GO Tn ? 7 CONTINUE *RITt(6,43)I I TS-!7 ?_l< <* CONTINUE ,  c c c c c  —  —  K=K+K  27  ,  T  3 8 ,  '  e N O  0  c c 2 0 9  1 0 1  • • I 2 , / / , 1 X I 1 2 5 ( 1H * ) t / i  THE S I M P L E X  ;  '  —  —  —  "~  "  —  CALCULATION  »HITE(6,209)  :  KF_AG»0 ' ' * ' '' ROUTINE' • / I T 4 9 I 2 6 ( 1 H * ) I / / / ) REAU.tbtlOl ) M P , KF.KM.(M_R( J),J»1,KM) FUKMA1 ( j I b , 6 F ] o , 0 ) "—' »R IT E I b, 1 0 1 ) M P • K F K M , ( Mn'R ( j ) , j o j ,KM ) AA=1,0/(KF*(SORT(2,0))) D = S()R'T ( K F + 1 . o ) O -AA*(H»l,0) ""— P'1-AA* ( ( K F - 1 , 0 ) +B ) T  —  ROUTINE  IFIKLlK.fcO.OIGO TO 1 0 0 ENC = 0 N = N-1 IF' ( K L I K . f c Q , 1 ) GO TO 1 0 0 GO TO 1 1 0 CONTINUE I F ( K O U N T , G E , 1 ) GO TO 2 8 5 SPENDLAYS  :  •  NUMBER  USING  ~  ~~—~  -  CYCLE  F  __i___N. I T T E R A T I O N ROUTINE  1 00  •  ~  1  ,  1  ,  T  9  ,  S  l  ,  P  L  E  X  ,  T  E  H  A  T  ,  0  •  N  t  M  \  —CALCULATE  c  2 1 0  21 1 2 6 2  -  c c c c c c  MATRIX  K»l DO 2 1 0 J ' l , K M | 2 WT S ( K ) 1 J 1 K=K + 1 K*l DO 2 1 1 J = 1 , K F RUM* ( J ) E H J R I K M ) K=K+2 FORMAT(3F20,0//I  —  CONSTANTS  — - MwR < K )  C A L C V L A TJL, .THE R A N G E  COVERED  — bY E A C H  G E N E R A T E THE M A T R I X I N THE FORM G E N E R A T I O N OF I M F lp. TFDMS TOT»KF»(KF+l)  •——  OF  MOLECULAR  A LINEAR  WEIGHT SET  VECTOR  —  KFR=KF+1  J»l 00  212  K»KFR|TQTtKFR  y » T S ( K )  C  212  J  »  J + 1  3  R Q M W t  GENERATION  C  J )  • P M  + »TS.  ,|)  • O F A L U T H E IQI  TERMS  KFEaKF+2 K F N = K F - i  KFEE=KFE+KFN  __.2 00 DO  2Bb NA" 1 iKFN 21_ J=KFfc,KFEE  * T S ( J )  a  RONiik  ( K ) *0M +»TS ( K )  IF IK . E O . (KF+1 ) ) K = l 213 C O N T I N iE KFKL=KFt E+KFR K F L * K F f - +KF'R K * K +1 285 CONTINUE IF lKFLAG , E 0 , 5) GO T O 2 4 0 |F (KFLAG . f c O . 4) G O TO 2 - 1 7 IF (KFLAG , E 0 . 3) G O T O 2 2 9 IF (KFLAG , L 0 , 2 ) GO T U 2 2 6 I F < K F L AG . E O . 1 ) GO TO 2 1 7 - J J L l K F L A G . F:0 . f, ) TO pQ*. i r K U A U , to . 7 G o To 2 9 7 .EST =0 *ORST=1000000 N_XT=0 L  •  ~  (  C C  FIRST  218  K T- 0 JC= 1 K= 1  __0 216  217  o  c  TO G E N E R A T E THE  2 1 b JBJCi  F-TESTS  K7  . L t . K F ) GO T O  ->|n  ORDERFIND THE OEST A N D WORST F I T CALCULATES THE LOCATION OF E A C H F-TEST 2S1  280  —>  REGRESSIONS  M»X(Kll| B W T S ( J ) K = K+ 1 KFLAfisJ -K.aU!iT = K ( ) U N T + 1 RE V IN0 l b . GO T O 1 0 2 KT"KT+l »R I J F ( f, , 2 7 4 > 4 N S » (1(1) FTLSr(KT) = ANSW(IO) JC«JC+KF KZ = K_ + K F  l* <-~T„  C C  c  FOUR  KZ = KF  DO 2 6 0 J = 1 , K F R IF(FTfcST(J) , L T , (OEST-1,0)) BEST=FTEST(J) ._Qf_.ua j CON TINuE DU 2 1 9 J = 1 , K F R  CO TO 2 8 0  IN  RELATION  TO T H E V E C T O R  —  —  V  ;  K  -\  V '  c c  219  IFIFTfcSTCJ) , G T . U O R S T + l , 0 t ) WORST « F T E S T ( J ) LOCKCJ CONTINUE  C-0 T b it 1 9  —  CALCULATE AVERAGE VALUE F O R N E X T NtXT*0 K=1 00 _ 2 0 J = l , K f - ' P IF(J , K 0 . L O C W ) G O TO 2 2 0 IF(J , E 0 , L O C H ) GO T O 2 2 0 _NEXr=FTKST(Jl+NEXT 220 CONTINUE' NEXT-NEXT/<KFR-2) ftHlTE(6.28l) PFST. fcORSTi NEXT #* _ _ M _ J _ Q H !* A J_ L__Ii! E T ' 1 F 1 S . 0 , 2 X . I * 0 R S T » , F 1 5 . 0 . 2 X . 1 N E X T 1 . F 1 5 . 0 ) L c CALCULATE CENTROID  —  _  —  —  —  —  L  S  c  P2«  923  KP = K F K««=LOCKI*KF KH=Kr KEND=T0T ClKK|S|) 00 2 2 3 K = K P » K F N D I K F 1F(K , E 0 . K » | GO T O 2 2 3 CIKX) o * T S ( K ) +C(KK) CONTINUE  —  "  C ( K K ) =C < K n ) / K F KW o K w - l KKtKK-| rKSPt - l-NKU D—-l\1fcPJ U • 1 IF ( K K , N t , 0 ) G O TO 2 2 4  c c c  -CALCULATE  P25  —  —  "  — —  '  —  "—"  —  :  T H F RFFIFCTmN  KW=LOCw*KF J - K F CONTINUE M».X(jtl) » C ( J ) + ( C ( J ) - t t ' T S ( K l * ) ) T M ' t X l J i l ) » M W X ( J» 1) J°J-1 K*= K w- 1 i r i o , N t , O i GO TO 2 2 5 KFLAG 2 GO TO 2'J9 > ± ' . ( _ A N S W ( 1 0 ) . G E . ( B E S T * i - n ) ) G O TO 2 2 7 .Gt. (NEXT + 1,0)) G O T O 233 GO TO 2 3 8 ' ' " » •0 <> T O 2 3 3  "  —  a  2g6  6  O  c c c  CALCULATION 227 228  E  ,  0  B  S  T+  1  1  0  O F EXPANSION  ~  '  —  REFLEC - ANSw(lO) .K*=LOC**KF  J*r.tCONTINUE M*X( j , l ) - C < J ) + 2*( ( C (J ) _ * T S ( K < * ) ) ) - J H. J__i X K*=K*-1 ~~ ' I F ( J , N E , 0 ) G O T O 228  .  KFLAG 3 GO T O 2 9 9 2 2 9 IF(ANS»( 10) B  ^ ~C  SUBSTITUTE 234  REFLECTION  FTEST(LOCV) •vUR_T=RE.FL_C K ». = L 0 C * * K F  GO TO 2 3 1  F O B WORST  AND CALC  NE«  CENTR0I0  HEFL6C  a  w I S< K W ) « r M » x ( J i 1 )  230  __.__!__. K 'ft K A =  1  I F ( J . , 0 ) GO T O 2 3 0 GO T O 2 S 1 F T £ _ T t L P C * I ' A H S o ( 1 fl ) VfO"«"S~T ° A N S * I 1 0 )  331  c C  , G E . REFLEC)  S U B S T I TUTE  EXPANSION  F O R WORST  AND CAUC  N F *  CENTROID  K'* = L O C * * K F J = Kf2 3 2  I * T S ( > C I F I ) - M A X < J , 1 )  _________ IF(J.NE.O) TO  I f  EXPANSION  M  _  £ 233  GO T O 2 3 2  2bl  GO  _____  RFFLECaANSw  IS  N O T A S GOOD  •  A S R E F L f c C T I ON GO B A C K  TO  ( 1o )  —GQ--XQ__3_  C  C  CALCULATION 235  K " - L Q .C._J_Ki: WtFi_fcC = A N S *  236  CONT I N|jt'  J  =  FOR CONTRACTION  .  OF  THE REFLECTION  io)  K F  _M»X ( J» l ) • C C J ) + . 5 » < . C l J ) - W T S ( K v n ) . J »  J  -  1  IF(J,NE,0)GO  C C  ('"• ^  TO 2 3 6  _____A__L__ GO T O 2 9 9  ?37  CHECK TO S E E I F IF(ANbiV(lQ) , LE , GO T O 2 9 8  CONTRACTION FIALED ( R E FL E C+ ) , 0 ) ) GO TO ? 9 A  C  CALCULATE  C  C  238  CONTRACTION  CLOSEST  TO T H E WORST  K * O L 0 C * * K F RLFLEC*ANS*(1o) J K F 3  2_U_-C_.NI.INU_  M * X ( j , i ) » =K A —1  C(J)-,_»((C{J)»WTSlK  K A J  S  J  -  1  J J L U . ^ E . O ) K F L A G« _ GO T O 2 9 9  G O TP  __9  *•)))"  LOCATION  HEPLECTION  C  \ '  c c  IF  CONTRACTION  CLOSER  TO  *0RST  _2AXL_LF(ANS .9J...LE. CW O R S T - I . O M 298 FTESTILOC*) • ANS*(10) W O « S T » F TEST ILOCW) Kvv = LOC**KF J«KF 241 CONTINUE WTS(K*) • MWX(J,1) J L U  V •  J3 J - J K '.V = K n - 1  IF(J.Nfc.O) GO TO 2 5 1  c c  PASSIVE  294 290  GO  TO  FAILS  GO  TO  295 — — "  —  241  H F F L F f r i n N  —  ~~  —  -  —  —  CONTRACTION  K * = LUC*'*KF J = KF CONTINUE »«Aij,i)-C(J)+ _5M(C(Jl.ttTS<K*)>> j a j - l Kw=K*"l J F (J.Nf-.0 )6Q T O 290  —  a  GU  c c  TO  —  —  299  M_L__i__E  295  —  WORST  rilNTUAi-TinN  Kw»LOC**KF J=KF  -2.9- CONT I -UE  297 2Hb  p»« * U , l | J I . , a b » U B S | (C<J>-*TS<KW) ) )) KW«K*-1 I f ( J . N f c . O ) GO TO 292 KFLAG=fo GO TO 2 9 9 I F ( ANSw< 1 0 ) , L T , J R E F L f c C l ,0> ) GO TO 2 7 0  GO  TO 2<ja  cc c c  —  1 r ( A N . * ( 1 0 I , L E . ( W O R S T - 1 . 0 > ) GO TO 2 7 0 GO TO 29B 2 7 0 * » l T r ( 6 , 2 7l) 2__L-£OR^ T_ ///, 45 I0ME OF T H F M A S S I V E C O N T R A C T I O N S i j a i l H ' l l . GO TO 110 299 KOUNT=KOUNT+l 1.FLKOUNT.GT.MDI « n TO I i n *R I T r (ft« 2 0 6 ) K O U N T 286 FORMAT</. II T T E R A T l O N NUMBER'|J4) nf> I T E ( b , 2 7 4 1 ANSW ( 1 0 ) _ t _ A _ F _ O R M A r ( / I F - T E S T 1 . F ? n , -n REMIND 16 GO TO 102 A  r w  —  L  T  >  ~ ~  ~  F A IL F n • . / . T4S . —  — ' — "  —  —  — —  —  • BEGINING  OF  THE  S M O O T H N I NG  ROUT! NE  se\?  uiR I T K fi.QOO 1  999  FORMAT( 1X,/,T58,•SMOOTHNING ROUTINE' i / i T 5 A . 1 8 ( READ(-•200) KA 1 KG,(DUMMY(J)•J»1tKG)  ~  1H * ) • / )  —  —  "  —  200 f~)  FOHMM ( 2 I 3 . 4 0 F 3 . 0 ) WRITE(6.903) K A . K G . <DUM*<Y(J  "^•C^'v-SilT^  K¥ = « N t l | » P * I REMIND 16 RF.AD<16.16> ( K IJMATt K >t K - 1 KB=1  KI) =  c c  B  | K G0 )  1  '  D  l  ,  M  W  V A R I A B L E S ' «I 31 2X t  T  iKV )  P-KA  H-V.F.  C  ) | J  LOAPl-D  KI.IMAT  WITH  »HAT  WAS  OR I fi I N A l  t  y  IN  THE  M l  F  KE=P+KD ft OO  ____.f_ KF =KE J = l DO 3 0 0 K»1iKC _____ O0_Ki_J_ij_A. K I J M A 1 ( K F ) - O U M M V ( J ) KF =KF t1 . " J t 1  __)j_XQNiaj____ 300 C C  KF=KF-KA KF = K F t P CONTINUE  DO  LOOP  300  KEEPS  TRACK  OF  THE  NUMB  OF  K I J  PER  C/CO  KB=Kfi+1  KE=KFtP 204  I F ( K B ,GT. i n 4 0p CONTINUE R E w1 N O lb  GO  I )  TO  204  GO  * R I T f c ( l 6 H 6 ) ( K l J M A T ( K ) , K " l f K Y ) GO T O 2 J _  C C C  SCHLIEREN 700  ROUTINE  (FORMULA  CHANGE)  WHEN  DATA(«tl)«0  CONTINUE  i i-^-jL'  AJL L « GO TO 701 STOP END S U B R O U T I N E Rt T URN END A  110  OPTICS  1  •  K  DATA  P  ,  * *  2  ,  *  (  *  W  X  (  J . K ) * * 2 ) * D E X P ( _ 0 A T A (  1,KP)*MWX(  J , K > «  L i s t i n g of the p o r t i o n of the S c i e n t i f i c Subroutine Package used f o r the MWD  program  C DECK  _£ c c c  J:  C O R R  0  0~ ,CORR 30 CORR S U B R O U T I N E C O R R E 40 CORR 50 -CORR P U R P O S E = — 60 CORR C O M P U T E M E A N S , S T A N D A R D D E V I A T I O N S , S U M S O F C R O S S - P R O D U C T S 70 CORR OF D E V I A T I O N S , A N D C O R R E L A T I O N C O E F F I C I E N T S , CORR SO __CQRR _ ?o USAGE C O R R I 00 CALL C O R R E < N , M . I O , X , X B A R , S T O , R X , R , _ , D , T ) CORR 1 10 CORR 120 DESCRIPTION OF P A R A M E T E R S C O R R _130 N - N U M H F H OF O B S E R V A T I O N S , N M U S T H E > OR B TO 2. CORR 1 40 M N U M B E R OF V A R I A B L E S , M r.^UST B E > OR TO 1 , CORR 150 IO - O P T I O N CODE FOR INPUT DATA C O R S 160 J2 I F D A T A A R E TO QE R E A D I N F R 0 M I N P U T DEVICE I N THECORR 170 SPECIAL S U B R O U T I N E NAMED DATA, ( S E E S U B R O U T I N E S C O R R ""180 U S E D BY T H I S S U B R O U T I N E B E L C " • ) CORR 190 1 IF A L L DATA ARE A L R E A D Y IN CORE, C O R R 200 XE—LOJLQJ) I _ _ V_AL.UE _ O F _ X J 5_0 , 0_» _ C O R R 210 I F 10=1, x i s T H E I N P U T M A T R I X ( N B Y M) C O N T A I N I N G C O R R ""220 DATA, C O R R 230 X B A R OUTPUT OF L E N G T H V E C T O R CONTAINING M E A N S i C O R R 240 S T D O U T P U T _.f c i O R OF LENGTH CPNTAINING S T A N D A R D _CORR 2 5 0 D E V I AT I O N S . C O R R 260 RX O U T P U T M A T R I X (M X M) C O N T A I N I N G S U M S O F C R O S S C O R R 270 P R O D U C T S OF D E V I A T I O N S FROM MEANS, C O R R 260 0J2IPUI_MATRix [ONLY UPPER T R I A N G U L A R P O R T I O N O F T H E _CORR 29 0 SYMMETRIC MATRIX OF »• l l v M ) c 0 N T A I N T N G CO R~R E L A T I O N C O R R "3 00 COEFFICIENTS. ( S T O R A G E MQDfc OF 1) CORR 3 t 0 OUTPUT V E C T O R OF L E N G T H M C O N T A I N I N G T H E D I A G O N A L C O R R 320 OF T H E u A „ l P J . X _ g X ^ U J l S _ O F _ _ C _ R Q_S S - P R O D U C T S OF _CORR_ J J O D E V I A T I O N S F R O M M E A N S . C O R R 340 D W O R K I N G V E C T O R OF L E N G T H M C O R R 350 T W O R K I N G V E C T O R O F L E N G T H M CORR 360 _C.QRR. R E M A R K S C O R R 380 C O R R E H I L L NOT A C C E P T A C O N S T A N T VECTOR, CORR 3 9 0 C O R R 400 S U ^ R U U T I N F S ANG F .UAX.IJLQ--.S-.. RH.OS RA M.S__R LQ UJLP E12 _ £ O R R _ 4J0 O A T A ( M ^ O ) T H I S S U B R O U T I N E MUST ME P R O V I D E D B Y T H E U S E R , C O R R 4 2 0" ( l ) I F io=o» T H I S S U B R O U T I N E I S E X P E C T E D TO C O R R 430 F U R N I S H A N O B S E R V A T I O N I N V E C T O R D F R O M AN C O R R 440 j£_iERNAi_JLNPJJT D E V , J C E C O R R _ 5 0_ ( 2 ) I F 1 0 = 1 , T H I S S U B R O U T I N E I S NOT U S E D BY C O R R 460 C O R R E B U T MUST E X I S T IN J O - DECK. I F USER C O R R 470 H A S NOT S U P P L I E D A S U B R O U T I N E NAMED D A T A t C O R R 480 T H E F O IJ-.Q " I M S I S S U G G E S T E D . C ORR 490 SUBROUTINE DATA C O R R 500 RETURN C O R R 510 END C O R R 520 - C O R R bJQ M E T H O D C O R R 540 P R O D U C T - M O M E N T C O R R E L A T I O N C O E F F I C I E N T S A R E C O M P U T E D , C O R R 550 C O R R 560 5J.Q ,-C.QR EL_: C O R R 580 S U B R O U T I N E C O R R E ( N , M , I 0 , X , X B A R , S T D , R X , H . B , D , T ) C O R R 590 —  m  Ul  X i n i X a A B l l ) l 5 T D l l ) . B < l l ) t m i ) . B | l l i D m i T l l l  MEN.I ON  IIMI C C C  If- A O C IN C S T A T E M D O U B L E P R  O U B L E P R E C I S I O N V E R S I O N O F T H I S R O U T I N E IS O L U M N i SHOULD B E REMOVED FROM T H E R O U B L E E N T .VHICH F O L L O W S . E C I S I O N XHAH.STn.RX.RiB . T i X  THE C MUST ALSO BE APPfcARING I N OTHER ROUT I N E ,  D E S I R E D . P R E C I S I O N  THE  REMOVED FROM D O U B L E P R E C I S I O N S T A T E M E N T S R O U T I N E S USED IN C O N J U N C T I O N WITH T H I S  THE D O U B L E P R E C I S I O N V E R S I O N O F T H I S S U B R O U T I N E MUST A L S O _C D - N J A I N DQIJBJLE P R E C I S I O N F O R T R A N F U N C T I O N S . SORT A N D A B S IN S T A T E M E N T 2 2 0 MUST B E C H A N G E D TO D S Q R T A N D D A B S , *  I N I T I A L I Z A T I O N DO 100  100  j a 1  F  M  t  K  .3.LJ ...= 0 . 0 T ( J | « 0 , 0 DO 102 I=l R III'Oill FNCN L » 0 L-110) DATA  105  127t  t  A R E A L R E A D Y  105  D0_a_a_._____l__-  107  T ( J )= T ( J ) + X ( L ) XB A R I J ) s T 1 J ) T ( J )= T ( J ) / F N  108  DO  107  lib  108 IN  CORE  )+0(J)*0(K)  R E A D O A S E R V A T I O N S A N D CALCULATE M E A N S F R O M T H E S E D A T A IN T U J  130  KK»N GO T O  3 0 .  137  130.  640 bbO 660  CORR CORR CORR  6 70 690 700  _2J0  _CORR_ 7 20 CORR 730 CORR 740 CORR 7 b0 _C.ORR C O R R "7 6 0" 770 CORR 780 ,CORR _CDRR_ 7 9 0 CORR CORR 810 CORR 820 _CORR_ 8 3 0 CORR 840 CURR 850 CORR 860 C C RR 870 CCRR 880 CORR 890 CORR 900  9 3 0  _CflRRI 020 CORRI 030 CORR i 0 4 0 CORR1050 _C0RR.0<S0 CORR10 7 0 CORR1080 CORR1090 _£0RR1100  I. ) - T ( J ) J ) +0 ( J ) J a 1 , M K M . J  1 27 I F . N . 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CORR 1 1 9 0 CORR1200  2-0 DO  2 3 0 J«liM K = J , M ~ JK=J+(K*K-K)/a L " M * ( J - I )+ K NX(L)= R(JK) _ =M * ( K - 1 ) + J RXIL)= R(JK) I F ( S T D < J ) * S T 0 ( K )) 2 2 3 , 2 2 2 , 2 2 5 Rl JK1=0.0 GO TO 2 3 0 R < J K ) » R ( J K ) / ( S T O ( J )*STD(K ) >  222  c c ccc c  225 230  240  r  STANDARD  —  FN»SQR1(FN-1,o) 0 0 240 J « 1 , M STD(JJ=STD(J)/FN I  HE  A  M  A  T  R  I  X  0  F  S  U  M  S  0  F  CROSS-PRODUCTS  Of  L " - M 0 0  2 5 0  I » 1 , M  1"L+M+1 B( 1 )sR X(L) R E T U R N END  DECK  ORDE  CORR1890 CORR 1 90 0 CORR1910 CORR1920 CORR1930 CORR1940 C O R R 19 5 0 CORR1960 CORR1970 C OR R 1 9 8 0 C0RR1990 CORH2000 C0RR2010 CORR2020 C0RR20J0 CORR2040 CORR2050 COHR2060 CORR2070 CORR2080 CORR2090 CDRRPlOn C0RR2110 C0RR2120 C0RR2130  ORDE • ORDE ORDE S U B R O U T I N E ORDER _0«DE ORDE PURPOSE ORDE ORDE C °£MI2UST -* MATRIX OF C O R R E L A T I O N COEFFICIENTS _0R0fc_ A SUBSET MATRIX OF I N T E R C Q R R E L A T I (INS AMONG INDEPENDENT ORDE w?_}?r!- ! 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U L I J.3 9 0 _ MULT 1 4 0 0 M U L T 14 10 MULT 1 4 2 0  —MULT 1 430_  125  S T A N D A R D  mo_  ____JL.T  MULT 1 3 6 0 MULT 1 3 7 0 MULT 1 3 8 0  00 .130 J = l iK Ll*KJJ_Jjj_Lt_l_ L»I S A V E ( J I  COMPUTED  ULT1160 MULT 1 1 7 0 MULT 1 1 8 0 _ j M U J _ . T _ l . i 9 0_ MULT1200 MULT12I0 MULT 1 2 2 0 MULT 1 2 3 0 MULT 1 . 4 0 MULT 1 2 5 0 MULT 1 2 6 0 -_MUL.T_1. _ / _ 0 _ MULT 1 2 8 0 MULT 1 2 9 0 MULT 1 3 0 0 M  ESTIMATE  >I*SY))  MULT 1 4 4 0 MULT 1 4 5 0 MULT 1 4 6 0 M U LT 1 4 7 0 _ MULT 1 4 8 0 MULT 1 4 y 0 MULT 1 5 0 0 M U L T 1 5 1 0__ MULT 1 5 2 0 MULT 1 5 3 0 MULT 1 5 4 0 _ M U L T 15 5 0 MUL T 1 5 6 0 MULT 1 5 7 0 MUL T 1 5 8 0  _MUL.T159Q__  MULT 1 6 Q 0 MULT 1 6 1 0 MULT 1 6 2 0 _ t _ U L I . l 6 3 0__ MULT 1 6 4 0 MULT1650 MULT 1 6 6 0 ____. I l b 7 0 _ MULT 1 6 8 0 MULT 1 6 9 0 MULf1700 _-__Jt-.T_L _l___ MULT 1 7 2 0 MULT1730 7  -J___JiY  iMINV MINV  10 20 30  VD  SUBROUTINE PURPOSE INVERT  A  U S A G E C A L L  TV  MINV  M I N V  MATRIX  ( A ,N,D  ,L. , M )  D E S C R I P T I O N OF P A R A M E T E R S A - INPUT MATRIX, DESTROYED RESULTANT INVERSE, N - f l R D F R OF M A T R I X A 0 - RESULTANT DETFRMJNANT L - WORK. V E C T O R OF LENGTH M • *'ORK V E C T O R OF LENGTH REMARKS MATRI X  A  SUBROUTINES NONE  MUST AND  UE  A  GENERAL  FUNCTION  :  IN  COMPUTATION  AND  REPLACED  BY  :  MATRIX  SUBPROGRAMS  REQUIRED  METHOD __—THE... S T A N D A R D G A U S S - J O R D A N M E T H O D I S U S E D . I S A L S O C A L C U L A T E D , A D E T E R M I N A N T OF Z E R O THE MATRIX IS S I N G U L A R ,  THE DETERMINANT INDICATES THAT  SUBROUTINE MINV(A»N,0•L»M) DIMENSION A( l ) , L ( 1 I,M( t )  IF A DOUBLE P R E C I S I O N V E R S I O N OF T H I S R O U T I N E I S D F S I R f c D , C 1N C O L U M N 1 SHOULD BE REMOVED FROM THF D O U B L E PRECISION STATEMENT WHICH FOLLOWS. DOUBLE PRECISION A iD •b IG A tHOLD THE C MUST A L S O B E APPEARING IN OTHER ROUTINE,  THE  RFMOVFD FROM DOUBLE P R E C I S I O N S T A T E M E N T S R O U T I N E S U S E D IN C O N J U N C T I O N WITH THIS  THE D O U B L E P R E C I S I O N V E R S I O N OF T H I S S U B R O U T I N E MUST A L S O CONTAIN DOUBLE P R E C I S I O N FORTRAN FUNCTIONS, ABS IN S T A T E M E N T 10 MUST B E C H A N G E D TO D A B S ,  S E A R C H  F O R  0 = 1 .0 N K » - N  DO _NK  55  ao K»1«N N K t__  L ( K ) = K  » ( K ) SK K K « N K + K  BIGA=A(XK• DO 2 0 J i K » N IZ< N * i J - 1 )  L A R G E S T  E L E M E N T  50 MINV 60 MINV M INV 70 MINV 80 MINV 90 MINV 100 ___._-_3___JU_ MINV 120 MINV 1JO MINV 140 __INV_150 MINV 160 MINV 170 MINV 180 MINV 19 0 MINV 2 0 0 M I N V 2 10 MINV 220 _MIN_V_ 2 3 0 M I N V ~2 4 6 MINV 250 MINV 260 _MINV_ 2 7 0 MINV 280 MINV 290 MINV 300 i__i.NV._JI5 MINV J 2 0 MINV J J O MINV 3 4 0 J-.I_N.__J5 0 i M I NV JfaO MINV J 7 0 MINV J 8 0 MINV 390 MINV A00 M I NV 4 1 0 _MI._LV_ _ _ J 0 MINV 4 4 0 MINV 450 M INV 4 6 0 __. I NV__ 4 7 0 M I N V 4 80 M INV 490 M INV 5 0 0 M I N V 5 10  , MINV 520 MINV 5 3 0 MINV 540 _MINV__50 MINV 5b0 MINV 6 7 0 MINV 5 8 0 -MJ_Ny_ 5 9 0 MINV 6 00 MINV 6 1 0 MINV 6 2 0 M I N V _____ MINV 6 4 0 MINV 6 b 0  10 IS 20 C  25  30  3B  40  ROWS  COLUMNS  I-If. ( K ) IF(I-K) 45.45.3H JP=N*( I- 1 ) OU 4 0 J = 1 i N JK=MK+J -LLiLJPjtJ HULD*-A(JK) A( JK ) = A ( J I ) A(JI) =HOLD D I V I D E C O L U M N BY M I N U S CONTAINED IN B I G A )  45 46 46 50 55  15*20.20  J=LIK) .IF(J-K) 35.35.25 K I = K-N DO 3 0 1 = 1 i N K I =K1+N HOID=-A {Kl | J I -K. I « K + J A(K1 ) 5 . ( J1 ) A(JI) "HOLD INTERCHANGE  35  MINV 6 6 0 MINV MINV MINV  CONTINUE 1N.ERCHANGE  c  c c  00 20 I«K.N IJ=lZ+l IF(DABS(QIQA)i 'DABS(A(IJ))) BIGAaA.IJ) L ( K ) Ml  I F ( B I G A ) 40.4ft.aB 0=0,0 RE TURN 00 65 1 = 1 § N IL± 1 -J____5j___LSu J3_CL. IK=NK+I A( I K ) = A ( I K ) / ( .. B I G A )  CONTINUE"  RfcUUCE  MATRIX  DO b 5 I•1•N IK =NK t I HUL D * A( I K ) IJ = I - N DO 6 5 J = 1 « N -XJ__I.__tN_ I F ( I - K ) bO , 6 5 . 6 0 60 IF(J-K) 62,65.62 62 KJMj-l+K A( I J ) = H O L D * A I K J ) + A ( I J ) 65 CONTINUE DIVIDE  ROW  KJ«K-N DO 7 5 J>liN  BY  PIVOT  PIVOT  (VALUE  OF  PIVOT  ELEMENT  IS  670 680 690 700 710 720 _7JQ_ 74 0 750 76 0 _7 7 0_ 780 790 800 _ 8 i o_ MINV B 2 0 MINV 8 3 0 MINV 8 4 0 MINV 8 5 0 MINV 8 6 0 MINV 8 7 0 MINV 8 8 0 _ _ l N V _ 8 9 0_ MINV 9 0 0 MINV 9 1 0 MINV 9 2 0 M I NV 9 3 0 _ M I NV 9 4 0 MINV 9 5 0 MINV 9 6 0 MINV 9 7 0 M I N V 9 8 0" M INV 99 0 MINV1000 _M I NV 1 0 1 CL MINV1020 MINV10J0 MINV104 0 __M J N V 1 0 5 0_ MINV106 0 M INV 1 070 MINV1080 M I N V I 0 9 0_ M I N V 1 10 0 MINV1110 MINV1120 M I N V I 1 3 0_ MINV.140 MINV1150 MINVt160 _M_INV 1 1 7 0_ MINV1180 MINV1190 MINv1200  J_i.N.yj2io_ MINV1220 MINV1230 M I NV1240 -JVI.N.V_12 50 _L MINV126 MINV1270  H  1  VD  7  70 ™ f  P  ,  )  n  J ' ,  K  1  u  C  K  J  I  C  B  .  t  c  P R O D U C T  c  T  /  , .  ' "  O F  —  M } N V ! J ! § M I N V I 3 A O  .  P  C  R E P L A C E BO  C £  Mj.Ny l J 1 P_  — —  P I V O T S  nsnjBif. -°*Hl<>*  ^  M I N V 1 « ! 8 0 M I N V 1 2 9 0 M I N V 1 3 0 0  ACKjT-i.Kj!^,'??  P I V O T  U V  M^ 13bO... V  M I N V I I W O  R E C I P R O C A L  C O N T I N U E  =  FINAL  «0V»  A N D C O L U M N  J  I N T E R C H A N G E  K »N 1 00 K = ( K- 1 ) 1F(K) 150fIbOt105 t 0 5 _ IJ»LXKJ IF(I-K) I20il20»106 JOB J U » N » H » ] ! J R cN *( I-1 ) DCL-.1.1 Q _ J - M i N JK = j(j + j HOLD=A(JK) JI=JR+J ft_( JK. )=--&( J I ) 110 A(JI> =HOLD 120 J=M(K) IF(J-K) 100»100tl2b DO 1 3 0 I = 1,N ~  •  •  —  K 7 = KV + N,  MINVlf.00  J.L5 K J - J S ± J _  KKi  is.i  / I I I  .  •—  130 ilSi ZDW ,  ^  5  0  60 10 100 RETURN  C  ^  -  ^  •  MINV1 A 00 MINV1 A 10 MINV1420 M.I N y 1 4 3 0_ MI N V 1 4 4 0 M I NV I 4i>0 MINVlAbO H.I N V. 1 4 7 0_ MINV1480 MINVJ490 MlNVlbOO f* I NV. 1 = 1 Q_ M1NV1&20 MINV1S30 MINV1540 M J N.v 1 6 b 0_ MINV1560 MINVlt>70 MINV1580 B.I.N.V-1 b 9 0_  —  —  —  —  _ _ _ _  •.  "  NV  620  MJ..N .VL1 a J V_  KJKXltSg  MINVlb50 MINV1660  MM1_NV_16J^_ I N V 1 6 B 0  2 01.  G.  Listing  of  the  simplex  program  202.  0> LT  —  r_ I o at  3 tn Z < _ o at  or — _  a  _  —  2 -t  a z  a  S _£  ui z a ui  _  a -» z a  S.-H —  o » x m -)ui| 2 IS)  X  r- < a * z * O -  z - -I  o — x| — o a ui — r 2 — Ul  »-  2 U l 4|> — J J UJ i o .-a •  _i _> u  Ul  a —  n m t• -  s  — i » a 5  3-. o •IPr_ — i>- ruiru • ru _ , »o_ . — o  — IL l O  J. —  _-!_ <iul —at < 2 f r e r a _j a l — * u. at c ; a !  U!  -  o  _  u. exj — _t ra o — mi • — — o -•in o * o — —_. X — — ru • _ J _ : _ ozj  as 3H« «ru  a u. _: • h o  o  — o —  — a  +" • I X I it  || - 3 •- 4 _ | o Z 4 _> U . U 1X1 c _ •- at _  n >  4 ui d a * * « 3 4 4 a: o — —»u) ii n u. *41r_oa  II  a  _ — ui o r*  _td  O • o ru u. i  4 4 a a ui U) iz Ul 13  — — — ru 4 + ru * + z _: — 0 « DiO H O _ a j a ic x o  0 1.  a  —ru  — to rum  UIO  OOUllJU  —  X  +• z a » — ->  •a 3 LL O _: a n _r n U.I  -t -tl  ui X  s o lui »  _l _1 4 z Z o  +1  ru — ui T| h a . O x B a h xi ->a « - i  a  O  |U1 z X o a H  -J U  ni — u i (Ul -fj a. o — x x _ i|-> Z h _q _ II — -> • — 4 n x rw — hZ z xi ru ui a l - z a x x o ii u o o _rJ _ j . - O I O *_c— u x  I  ru —  -I  nc  I  n * — 1 _  njru  /~\  gab  KFEfc«KFEfc+KFR KFEcKFE+KFR K»K-f 1 C 0 (SI I IJ ME . IF (K UAG ,fcO, GO 1 0 240 IF (KFLAG . E Q , 4 1 GO T O 2 3 T IF IKFLAG , E 0 , 3 ) GO T O 2 2 9 IF (KHL.AG . FQ. 2 ) GO T O ? 2 b IF(KFLAG . E O . 1 ) GO 1 0 2 1 7 I F ( K F L A G . E O . 6 ) GO T O 2 9 9 IF(KFLAG,EQ.7)GO T O 29*, liEJii . 0 iOWST=100000000 N E. X T = 0  f  b)  :  g  —  C C C  FIRST  FOLIO.  K F GR E S S I ON 5  KP = KF K I =0  2ia  TO  GENERATE  THE  :  F-TFSTS  :  K = I  DO  3 1 b J=JCi  KP  KFLA6=1  fiQ_r.o_i.oi  C C C  gbl  280  -  C  219  c  220  •J -  c  C  c  : 218  O R D L R - F I N D T M F OfcST A N D WORST F I T C A L C U L A T E S T u t L O C A T I O N OF E A C H F - T E S T  C .  :  216 « » X ( K i | ) = ATS(J> 2 l 7 K T = K T t l F T t S T U T ) » ANSW (10) JXgjlC t KF KP a K P + KF IF ( K T , L E , K F ) GO T O  201  I N RtLAT I O N TO  THE  280 J=1»KFR L t l F I F- S T ( J ) . L T . B E S T ) GO T O ; B E S T =F T E S T < J > LOCfci = J CONTINUE B.Q 2 19 J = l . K F R ; lF(FTtST<J) , 0 T , W O R S T ) GO T O 219 C» iO ONRTSII N U»E F T E S T ( J > LOC*"J CALCULATE A V E R A G E VALUE F O R NEXT NfcXT=0 K =l DO 2 3 0 J « l i K F O I F ( J , £ Q , L O C A ) G O TO 2 ? O I F ( J . E O , L O C O ) GO T O 2 2 0 NEXT=FTEST(Jl+MFXT CUNT INUE NF:XT=^EX T/ (K F R - 2 ) * R I T F ( 6 i 2 7 4 ) ( F T F S T ( J ) •J=1 ,4) *BlTt(ni281l flF«iT«*np«-T i M F X T FURiiAT(«riESri,2XiFlb.5,i*0RST-«2X,Flb,b»INEXT.|2X|F15,5) DO  CALCULATE  VECTOR  280  ro  o  CO  •  CENTR01D  KP=KF KwaLOCiiK*KF  A  204.  ft)  MM] ftl  o ftl  Ci otI-  (0  U-  »»>  ftl ftl — X. o x  *  MO• * i l  X  *  — D a  ix l ul a o o zn t- II »> « ~ I \ I | X X *- ¥ O X . M |- K Z J : z < — o — ' UJ — O — a o a x  «  ('  a  — X a  *  a it  >!zfti• » - X ,0x i x —>UJ «3 II o—— £ z ui u. a o x l — * u.  ui — — — ui 2 ftl 3— I •!! — -> — H f V3 >- — X I X - > < 1 Z X * T( II — J . o i s n < u. u. : XT o z o lx — x  u c  M  ftl  (  Ul  1-13  O  :  (  s  3 — JO O  »M ——I ^O— i »«i|/|  z o •nz <t — _ — < —I I O U i L . o —  -o  '  U.  I  n ftl fti  ftl ft!  (  x x u x. x  i a x n a x  _l  K X </>Ul UJZ  •J, — 3 o-> — 13— C a a — — io z ~ oi • 10 o  —  rxtt •-  I  13  c ->-  — X  o  oo|  — •o •»t»i • ft! « X ft: —fti n •  ftl ftl  M  u. o — v3  z  ui • 3-  : ro  x j o  -  u IZ I J «IO o _l j u j i I rx|-» < - I I I U — < a u.»iizx-i u-a: Ul * U I O t H a x - » o z ~> x i - x * » al  ftl  ro ftl  —  •  -> — X  *  « u.  M  —  a U — Ul 3 -1U. o — OU. X a —) • « 3 Ll * a _ .i a *, — I UJ ftl — MO! < <z t-r-d X — X| " O " ail/1 J i - I  B  Jt  o  o Ui _l u. ui a  uivm >  M  c U_ O tjor X j " J =c "» xj — 1 3 l — 3 *! B — M 1 ftl ftl  tti  231  c c  FTtSTILOCw)«ANS*110> KOHSTBANSV.'( 10)  SOOSTITUTE  E XP A N S TOM  KV<=LOC*»KF J = KF WTS(K*)=Mwx<J.M j a j - 1 K * *Kw™ 1 I F ( J . N E . O ) GO T O GO T O ? « . !  "  FOP  WORST  CALC  AMD  NEW  —  —  —  CWNTRniD  —  —  ;  232  c  c  IF _2_3  c c  CALCULATION 235  23b  1  EXPANSION  2/a  c c  IS  GOOD  AS  REFLECTION  CONTRACTION  OF  THE  NOT  AS  BACK  TO  REGLECTI ON '  FOR  K* =L O C » * K F R E F L t C = ANSwI 10) J»KF CONT I NIPE M * X ( j i l ) » P ( J )+ , 5 * ( 1 P I J ) . » T J =J - 1  —  REFLECTION —  —  S ( K * I  ) )  —  —  I F t J . N E . O I G O TO 236 Kr-LAG-4 * R 1 T E I 6 > 2 7 8 ) ( M w x ( J • ) ), j a 1 ,3> F O R M A T ! / , " C O N T R A C T I O N - R < , 2 X . 3 F 2 0 . 0 l GO ro 102  — — — :  —  CHECK TU S E E IF C O N T R A C T I O N FIALED IF(ANSVvIio) , l . T , REFLEC) GO TO 294 GO TO ?98  c c  CALCULATE ?  .8  CONTRACTION  CLOSEST  THE  TO  279  c c  LOCATION  "  J 3 K F  —  —  CONTlNuE M * * ( J . l ) = P l J l - . 5 * ( ( P l J l - * T S ( K W , | ) K » » K » - t J =J - 1 I F ( J . N E . O ) GO TO 239 KFLA_=H W R I T E ( 6 . 2 7 9 ) ( M W X ( J , 1 ) , J * l , 3 ) F O R M A T ( / . • C O N T R A C T I ON-». 1 , 2 X » 3 F 2 0 , 0 ) GO TO 102 IF  8*0 ?9fl  WORST  K'A = L 0 C * * K F  REFLEC«:ANS*(io> 239  o  _0  R E F L F C B A N S * ( i n ] GO TO 2 34  CONTRACTION  IF(ANSw(10) F T E S T l L O C * )  CLOSER  , L T , *ORST) a A N S M 10)  »ORST«FTfcST(LCVC») K* • LOC«i*KF  TO GO  WORST TO  FATLS  295  —  ~~  206.  JJ •  o ftl u.  o Ul  O ftf  3  tn z o  x  t ac i z —c  2 3  2 O  — OJ X > -( -  O  —in ~< —5 Cf -  V9  O — • j> I — J J I M •  X II s X  —  • o -> i — u a ii ij  cd  < cr z  JJ JJ £  I  »  Jt  CM  * I M  1- JI  — tn • Ui J I  • xzji - M 3 ivs z » a ~ 3 - x!in — o •—- »o H a x zt-i i 3ui — + 1 • OJ t - - cd-> •z ? t - J J x f-1 4 I O 03 O X - Hlrru"> — — o •f t i• ) — — X »- • \H • r*. u Z J J " " " ^ j * ™" I ui a I <t 3 t -o u i < — • £ t - Zx i - z i f - — a ool • or 3— - . n t n — 3 O O j|u- a 323ct o t - z ) * u . <1 < t3 W i i i u. ts x  0  x> s —« s tt —in — 4 Ot(V •  in ac  ~i -> > < 3 z ii — —! -< u z '•3 J J < c m 0 •-• JU.J in «x!i 4 i - i h JU.1II X z x - _l ~cr < II X z J u II t u . u. at 3 a r 1 n o ->XJ  u o  —  tn •a Z  I —z —a  3 UJ  Z 3 o x O—  t_i  ~>-t •I—u i<  z c  Z o  z a  I  Z —o  a. * Ul Z • X  -•in JJ I lit  •  X  o — 5) — « |  q —0" — a • H « < \ i * ni  I in • JJ— I VOIZ 4- *'* ->< c z n  x~xi  - r - U OJ < O i <  — -x  ac a a. * u. x  »»  CP  ct i <* •- <  _1  gi a>  JJ cr  ' I-  o — i^rrun.  1  1  at o »4  ft:  )  ;  

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