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UBC Theses and Dissertations

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

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EQUILIBRIUM ULTRACENTRIFUGATION AS A METHOD FOR STUDYING PROTEIN INTERACTIONS by F r e d j v ^ n ^ e V o o r t B.Sc.(Agr.), U n i v e r s i t y of B r i t i s h Columbia, 1972 M . S c , U n i v e r s i t y of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF FOOD SCIENCE UNIVERSITY OF BRITISH COLUMBIA We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1977 Fred van de Voort, 1977 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lumbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and stud y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f f^A c ^ ^ ^ The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date S e o T i l «n 7 ABSTRACT The u l t r a c e n t r i f u g e has t r a d i t i o n a l l y been used to o b t a i n m o l e c u l a r weights, b ut r e c e n t developments have expanded the c a p a b i l i t i e s of t h i s instrument. One of these developments was the a b i l i t y to o b t a i n the complete m o l e c u l a r weight d i s t r i b u t i o n (MWD) from u l t r a c e n t r i f u g a l data through the use of a l i n e a r programming c a l c u l a t i o n developed by S c h o l t e . One o b j e c t i v e o f t h i s r e s e a r c h was to apply the MWD c a l c u l a t i o n procedure of S c h o l t e t o the study o f p r o t e i n systems. In p r e p a r a t i o n f o r t h i s work, the u l t r a c e n t r i f u g e was equipped w i t h new UV o p t i c s and i n t e g r a t e d w i t h a data a c q u i s i t i o n system coupled w i t h a desktop computer to a l l o w the routine c a l c u l a t i o n o f m o l e c u l a r weights from u l t r a -c e n t r i f u g a l data of both homogeneous and heterogeneous systems. The combination of u l t r a c e n t r i f u g e and data a c q u i s i t i o n system was found t o work w e l l i n s p i t e o f 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 system. A new c a l c u l a t i o n procedure f o r o b t a i n i n g the MWD was a l s o i n v e s t i g a t e d . Development of the frequency v s . m o l e c u l a r weight f u n c t i o n was based on m u l t i p l e r e g r e s s i o n r a t h e r than l i n e a r programming and had the advantage of p r o v i d i n g s t a t i s t i c a l parameters to assess the accuracy o f the f i t t i n g procedures. A FORTRAN program was developed and t e s t e d i n regard to i t s c a p a b i l i t i e s and l i m i t a t i o n s , through the a n a l y s i s o f model systems. The m u l t i p l e r e g r e s s i o n approach was found t o be e q u i v a l e n t to the method of S c h o l t e . The a v a i l a b i l i t y o f the s t a t i s t i c a l parameters l e d to the i n c l u s i o n i n the MWD program of a simplex o p t i m i z a t i o n r o u t i n e t h a t allowed the e v a l u a t i o n of the weight average m o l e c u l a r weights and the c o n c e n t r a t i o n s o f i n d i v i d u a l components pr e s e n t i n multicomponent systems. Using the simplex r o u t i n e to search f o r the bes t f i t s o l u t i o n s t h e o r e t i c a l l y allowed a r e d u c t i o n i n the number of r o t o r speeds r e q u i r e d t o o b t a i n a d i s t r i b u t i o n , thus e l i m i n a t i n g one of the major r e s t r i c t i o n s of S c h o l t e ' s method. Attempts t o v e r i f y e x p e r i m e n t a l l y the t h e o r e t i c a l advantages of t h i s new approach o n l y p r o v i d e d an i n d i c a -t i o n t h a t the method was v i a b l e . Due t o the l i m i t a t i o n s of the a v a i l a b l e equipment, the MWD c a l c u l a t i o n c o u l d not be put t o use i n studying p r o t e i n systems, although the p o t e n t i a l f o r i t s use i n such work was demonstrated. i . c TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS X INTRODUCTION . . 1 LITERATURE REVIEW . 4 PART I . MOLECULAR WEIGHT DETERMINATIONS 12 THEORY . . 12 EXPERIMENTAL 18 U l t r a c e n t r i f u g a t i o n 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 system 2 5 . 2. Desktop computation and p l o t t i n g 28 3. A p p l i c a t i o n o f the data a c q u i s i t i o n system to the c e n t r i f u g e • - • • 30 B. Software 31 1. Data c o n v e r s i o n program 31 2. Automatic l i n e a r r e g r e s s i o n and p l o t t i n g program 32 3. Automatic m u l t i p l e r e g r e s s i o n program w i t h back, c a l c u l a t i o n f e a t u r e 32 RESULTS . . 3 3 Homogeneous Systems 33 i i . Page Heterogeneous Systems 40 DISCUSSION 49 PART I I . MOLECULAR WEIGHT DISTRIBUTIONS 53 THEORY 53 Linear Programming - The Solution of Scholte ... 5 8 Multiple Regression as an Alternative Solution 62 RESULTS AND DISCUSSION 68 Model Systems 6 8 A. Log normal d i s t r i b u t i o n method 69 B. Simple expansion of the Rinde equation 71 Molecular Weight D i s t r i b u t i o n Program Using Multiple Regression 77 Comparison of Linear Programming and Multiple Regression 91 Factors A f f e c t i n g the Molecular Weight D i s t r i b u t i o n 9 3 A. Speed 94 B. Interval and range 97 C. Loss of data 100 D. Summary 100 Expanding the Multiple Regression Approach 101 A. Assessing the s t a t i s t i c a l parameters 101 B. The concept of an i t e r a t i v e solution 102 C. The i n i t i a l i t e r a t i v e algorithm 106 i i i . Page D. The simplex i t e r a t i v e algorithm 10 8 1. The simplex method 109 2. The s t a r t i n g matrix 114 3. The simplex output 118 E. Smoothing of undefined data 123 Case Studies of Some Model Systems 126 A. Catalase 126 B. Trypsin inhibitor-ovalbumin-conalbumin 130 C. as^-casein and K-casein and in t e r a c t i o n product 132 D. Lysozyme-ovalbumiri and in t e r a c t i o n product ' 134 E. Discussion . 135 Analysis of Protein Mixtures 136 A. Trypsin inhibitor-conalbumin 138 B. Trypsin inhibitor-ovalbumin-conalbumin 139 C. Ovalbumin-thyroglobulin I 4 2 D. Catalase 1 4 2 E. Discussion I 4 4 Other Factors ^-4^ A. Extinction c o e f f i c i e n t s l 4 ^ B. P a r t i a l s p e c i f i c volume I 4 7 CONCLUSION - 1 5 0 LITERATURE CITED 1 5 7 i v . Page APPENDIX 164 A. Proteins Used i n This Investigation 164 B. Assembly of Yphantis C e l l 164 C. Programs for the Monroe Calculator 165 1. Time to reach equilibrium program 165 2. Orthogonal polynomial curve f i t t i n g program 168 3. Data conversion program for manually selected data 169 D. FORTRAN Programs 169 1. The molecular weight d i s t r i b u t i o n program 173 (a) Calculation mode 173 (b) I t e r a t i v e mode 174 (c) Smoothing mode , 175 2. The simplex program 176 E. L i s t i n g of the MWD Program 177 F. L i s t i n g of the Portion of the S c i e n t i f i c Subroutine Package Used for the MWD Program 18$ G. L i s t i n g of the Simplex Program 201 LIST OF TABLES Ta b l e Page I. M o l e c u l a r Weights Obtained f o r Ovalbumin 4 2 I I . M o l e c u l a r Weights Obtained f o r C a t a l a s e 50 I I I . Comparison of Frequencies R e s u l t i n g from A n a l y s i s of a Two Component System Using M u l t i p l e Regression and L i n e a r Programming .... 92 IV. F r e q u e n c i e s and F- R a t i o s Obtained by M u l t i p l e Regression A n a l y s i s of a 2 0 , 0 0 0 - 8 0 , 0 0 0 D a l t o n M i x t u r e (1 :1 Ratio) Using S e l e c t e d M o l e c u l a r Weight P a i r s - .105 V. Summary of the Simplex C o n t r a c t i o n O p e r a t i o n s . . 115 V I LIST OF FIGURES Figure Page 1. Schematic diagram of the UV o p t i c a l system . . . . 20 2. Recorder trace (scan) of a t y p i c a l sedimentation equilibrium experiment using a double sector c e l l . . . . 21 3. Photograph of the data a c q u i s i t i o n system on a laboratory cart - teletype, voltmeter and d i g i t i z e r 26 4. Photograph of the a c q u i s i t i o n system, multiplexing unit and u l t r a c e n t r i f u g e 27 5. Schematic diagram or the data processing operations 29 6. Schematic diagram of data flow through the data conversion, l i n e a r regression and multiple regression programs . . . 34 7. Facsimile of teletype output 36 8. Input and output of the data conversion program and l i n e a r regression plus p l o t t i n g program 38 2 9. Plot of In A vs. x for ovalbumin showing the le a s t squares f i t to the data 41 2 10. Plot of In A vs. x for catalase showing the l e a s t squares quadratic f i t to the data . . . 44 11. Input and output of the multiple regression program . . . . . . . . . 45 12. Input and output of the multiple regression back c a l c u l a t i o n „ 47 13. Input and output of the log normal d i s t r i b u t i o n program . . . . . 72 14. Semilogarithmic p l o t of the output, f(M) vs. M, from the log normal d i s t r i b u t i o n program for a three component system of 25,000-80,000-320,000 daltons (1:1:1 r a t i o ) . . . . 74 v i i . Figure Page 15. Semilogarithmic p l o t of f(M) vs. M f o r a 25,000-80,000 dalton mixture (1:1 r a t i o ) . Multispeed data; i n t e r v a l =2.0 74 16. Input and output of the £ vs. c(£)/c program for the log normal d i s t r i b u t i o n 75 17. Input and output of the Rinde equation program 78 18. Example of the output of the multiple regression MWD program . 8 4 19. Semilogarithmic p l o t of f(M) vs. M for a 25,000-80,000 dalton mixture (1:1 r a t i o ) . Single speed data; i n t e r v a l = 2.0 . . . . . . . . 96 20. Semilogarithmic plo t of f(M) vs. M for a 25,000-80,000-320,000 dalton mixture (1:1:1 r a t i o ) . Single speed data; i n t e r v a l = 2.0 96 21. Semilogarithmic p l o t of f(M) vs. M for a 25,000-80,000 dalton mixture (1:1 r a t i o ) . Multispeed data; i n t e r v a l =1.5 98 22. Semilogarithmic p l o t of f(M) vs. M for a 67,000-134,000 dalton mixture (2:1 r a t i o ) . Multispeed data; i n t e r v a l =2.0 . . . . 98 23. Semilogarithmic plo t of f(M) vs. M for a 67,000-134,000 dalton mixture (2:1 r a t i o ) . Multispeed data; i n t e r v a l =1.4 99 24. Semilogarithmic p l o t of f(M) vs. M for a 67,000-134,000 dalton mixture (2:1 r a t i o ) . Multispeed data; i n t e r v a l = 1.4, with the range having been adjusted 99 25. I l l u s t r a t i o n of the f i r s t basic operations of the simplex optimization routine I l l 26. Flowchart for the simplex algorithm 119 27. I n i t i a l output of the simplex algorithm 120 28. Semilogarithmic p l o t of f(M) vs. M for a 25,000-80,000 dalton mixture (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 125 F i g u r e v x n , Page 29. 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 s p e e d 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 30. 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 and 15,000 rpm, i n t e r v a l = 2 . 0 . . 126 31. 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 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 u n d e r g o n e s m o o t h i n g . . . . . . 126 32. 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 ) . 13,000 rpm, i n t e r v a l = 2.0 131 33. 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 = 2.0 131 34. 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 35. 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, 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 133 36. 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 and 13,000 rpm, i n t e r v a l = 2.2 137 37. 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 ) . 10,000 rpm, i n t e r v a l = 2.2 . . . 137 38. 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 c o n a l b u m i n . 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 and 21,600 rpm . . 140 39. 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 c o n a l b u m i n . 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 and 21,600 rpm, 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 140 i x . Figure Page 40. Semilogarithmic plo t of f(M) vs. M for a mixture of tr y p s i n i n h i b i t o r and conalbumin. Absorbance = 0.5 for both, mixed 1:1 by volume. Run at 19,600 rpm 141 41. Semilogarithmic plo t of f(M) vs. M for a mixture of tr y p s i n i n h i b i t o r , ovalbumin and conalbumin. Absorbance = 0.5 for a l l , mixed 1:1:1 by volume. Run at 9,600, 15,500 and 21,600 rpm 141 42. Semilogarithmic plo t of f(M) vs. M for a mixture of ovalbumin and thyroglobulin. Absorbance = 0.5 for both, mixed 1:1 by volume. Run at 6,300, 9,600 and 15,500 rpm . . 143 43. Semilogarithmic plo t of f(M) vs. M for catalase. Absorbance = 0.5, run at 9,600, 15,500 and 21,600 rpm 14 3 IA. Time to reach equilibrium program 166 2A. Input and output of the data conversion program for manually selected data 170 3A. I l l u s t r a t i o n of card deck for various modes of the MWD program and for the simplex program 172 ACKNOWLEDGMENT I wish to express my s i n c e r e g r a t i t u d e t o Dr. Shuryo Nakai f o r h i s a i d and encouragement throughout the course o f t h i s work. Working w i t h him has been a t r u e l e a r n i n g experience. I a l s o would l i k e t o thank Dr. M. Tung f o r h i s advi c e and l o a n o f computer d o l l a r s . My committee members were most h e l p f u l e d i t i n g the f i n a l t e x t , f o r which I am most g r a t e f u l . I would a l s o l i k e t o acknowledge the f i n a n c i a l a s s i s t a n c e which I, r e c e i v e d over the past t h r e e y e a r s i n the form of a Leonard S. K l i n c k F e l l o w s h i p . F i n a l l y , I would l i k e t o thank my w i f e , C o l l e e n , f o r her forbearance and he l p i n p r e p a r i n g t h i s manuscript which I d e d i c a t e to her. INTRODUCTION The o v e r a l l structure and properties of many food products are dependent on the interactions of three major components: proteins, l i p i d s and polysaccharides, usually as aqueous dispersions. S c i e n t i f i c study of food systems has led to the r e a l i z a t i o n that fundamental food components can be i s o l a t e d by processing various raw materials and l a t e r recombined into new food products. This approach to food manufacture requires an expanded base of fundamental knowledge that can be applied to the problems involved i n food f a b r i c a t i o n . In the case of proteins, there are numerous situ a t i o n s i n which t h e i r interactions are important to a food system. A c l a s s i c example i s the thinning of egg white during storage, which i s believed to be caused by the i n t e r -action 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 gnificance of protein-protein interactions include the i n t e r a c t i o n of k-casein with 6 -l a c t o g l o b u l i n r e l a t i v e to the heat s t a b i l i t y of milk, and the in t e r a c t i o n of p r o t e o l y t i c enzymes with t h e i r substrates. Since these interactions are an i n t e g r a l part of foods, the elucidation of the mechanisms and forces involved (such as hydrophobic and e l e c t r o s t a t i c ) are fundamental to the understanding of food systems. 2. However, the complexity of a food product n e c e s s i t a t e s t h a t i n i t i a l l y a more manageable system be s t u d i e d . P r e v i o u s work i n t h i s l a b o r a t o r y (9,10,27,34,35,36)* on macromolecular i n t e r a c t i o n s made use of c o n v e n t i o n a l methods such as g e l f i l t r a t i o n , v e l o c i t y u l t r a c e n t r i f u g a t i o n and f l u o r e s c e n c e p o l a r i z a t i o n . Each of these methods had a l i m i t a t i o n i n i t s a p p l i c a t i o n t o these s t u d i e s . In the f i r s t two methods, the Johnston-Ogston e f f e c t h i n d e r e d the e v a l u a t i o n of a c c u r a t e thermodynamic data, and i n the case o f f l u o r e s c e n c e p o l a r i z a t i o n , o n l y l i m i t e d i n f o r m a t i o n was p r o v i d e d . Sedimentation 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 i s another p o t e n t i a l method f o r the study of 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 . T h i s approach has the advantages o f b e i n g f i r m l y based on thermodynamic theory, and of a l l o w i n g the system under study to come to e q u i l i b r i u m . U n t i l r e c e n t l y , o n l y a l i m i t e d amount of i n f o r m a t i o n , such as the weight average or z average m o l e c u l a r weights c o u l d be o b t a i n e d through the use of t h i s technique (45). 1 T h i s l i m i t a t i o n was overcome i n 1968 when a p r a c t i c a l mathematical s o l u t i o n was found t o the Rinde e q u a t i o n (44), thereby a l l o w i n g the complete m o l e c u l a r weight d i s t r i b u t i o n (MWD) to be determined from sedimentation e q u i l i b r i u m d a t a . Thus, i t was f e l t t h a t t h i s approach c o u l d be a p p l i e d t o the * Numerals i n parentheses r e f e r t o the l i t e r a t u r e c i t e d . study of food proteins and t h e i r i n t e r a c t i o n s . This thesis focused on two major objectives. The i n i t i a l aim was to put into operation an u l t r a - v i o l e t (UV) o p t i c a l system newly acquired for our L2-65B preparative u l t r a centrifuge and to integrate t h i s unit with a data a c q u i s i t i o n system and a desktop computer. These components were obtained to allow for the automation of molecular weight determinations. The second objec-t i v e was to apply the MWD c a l c u l a t i o n to proteins, using model systems and protein mixtures to evaluate i t s p o t e n t i a l usefulness i n assessing component concentrations. LITERATURE REVIEW Many macromolecules such as synthetic polymers and biopolymers, are produced i n various chain lengths and are, therefore, heterogeneous i n molecular weight. The measurement of t h i s heterogeneity i s of fundamental importance i n macromolecular chemistry because the molecular weight d i s t r i b u t i o n of a polymer i s re l a t e d to i t s end use c h a r a c t e r i s t i c s or performance. A p r a c t i c a l example of such a rela t i o n s h i p has been provided by Gehatia and Wiff (21) who discussed how polymers made up of d i f f e r e n t MWDs could a f f e c t the performance of decelerator parachutes for space vehicles and high speed a i r c r a f t . In the case of biopolymers, the MWD of dextran was of in t e r e s t to Williams and Saunders (65,66) , since there appeared to be a close r e l a t i o n s h i p between the molecular size of the dextran molecules and th e i r retention time i n the body, when dextran was used as a plasma extender. Analysis of the MWDs of polymers i s not r e s t r i c t e d to determining end use c h a r a c t e r i s t i c s . Determination of MWDs could also be useful i n studies of macromolecular associations and interactions (2 4). Proteins, which are usually homogeneous, quite often s e l f - a s s o c i a t e and thereby become polydisperse i n terms of molecular weight. 5. Therefore, protein mixtures can be studied for macro-molecular interactions to elucidate the reaction mechanisms and to calculate the association constants (43). In order to evaluate the p o l y d i s p e r s i t y of a macro-molecular system, most methods either separate or re-d i s t r i b u t e the sample. In general, complete separation i s not achieved so that computational methods are required to elucidate the 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. The ultracentrifuge was one of the f i r s t instruments used to evaluate 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 equilibrium u l t r a -centrifugation has 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 foundation based on the p r i n c i p l e s of thermo-dynamics (19). In t h i s method of analysis, the macromolecules are placed i n a moderate c e n t r i f u g a l f i e l d forming a concentration gradient that i s measured o p t i c a l l y . From these data the molecular weight of an i d e a l macromolecule can be calculated using the following r e l a t i o n (11): 2RT din c M Eq. 1 app (1 - v p ) w 2 where: apparent molecular weight of the solute (daltons) R universal gas constant (8.314 x 10 ergs/ deg. - mole) 6. T - temperature (°K) P - density of the solvent (g/ml) v - p a r t i a l s p e c i f i c volume of the solute 0) - angular v e l o c i t y (rad/s) c - concentration of the solute (g/1) r - r a d i a l distance from center of ro t a t i o n (cm) For an homogeneous i d e a l solute, a p l o t of the natural logarithm of concentration versus r a d i a l distance squared w i l l give a st r a i g h t l i n e with the slope d i r e c t l y propor-t i o n a l to the molecular weight of the solute. In the case of i d e a l polydisperse systems of chemically i d e n t i c a l molecules, a c u r v i l i n e a r p l o t i s obtained. Various molecular weight averages such as the number average (M n), weight average (Mw) and z average (M^) can be calculated by several procedures ( 1 9 , 2 9 , 3 9 , 6 4 ) . These molecular weight averages give limited insight into the f r a c t i o n a l d i s t r i b u t i o n of the component molecules of the system. In p r i n c i p l e , sedimentation equilibrium u l t r a c e n t r i f u g a t i o n i s capable of providing the entire molecular weight d i s t r i b u t i o n , as well as the weight average molecular weights. 7. The p o s s i b i l i t y of obtaining the MWD from the concentration gradient formed by equilibrium u l t r a -centrifugation was pointed out by Rinde i n 1928 (44). He developed the following r e l a t i o n s h i p on which most of the methods for obtaining MWDs from u l t r a c e n t r i f u g a l data are based. c(£) f XM exp(-XM?) f(M)dM Eq. 2 c o 1 - exp(-XM) where: X - i s a complex function of rotor speed £ - a dimensionless r a d i a l coordinate f(M) - d i f f e r e n t i a l molecular weight d i s t r i b u t i o n c - i n i t i a l concentration o c(£) - concentration at some r a d i a l distance M - molecular weight (daltons) P r i o r to the completion of Rinde*s d i s s e r t a t i o n , Svedberg and Nichols (53) attempted to obtain a MWD using Rinde's i n t e g r a l r e l a t i o n . However, t h e i r approach, based on simultaneous equations, resulted i n a solu t i o n con-taining negative concentration values that were not r e a l i s t i c i n terms of the physical s i t u a t i o n . In 1935, Lansing and Kraemer (2 9) demonstrated how to obtain the molecular weight averages M^ and M . They showed that 8. the three molecular weight averages, Mn, M w and Mz, could be used to calculate the probable MWD, assuming that the MWD followed a log normal d i s t r i b u t i o n . This approach, which attempted to f i t molecular weight moments to an assumed d i s t r i b u t i o n , and other s i m i l a r approaches were l a t e r shown to be impractical (28,67). Wales and h i s associates (55-58) t r i e d to avoid the fixed functional forms of Lansing and Kraemer (29), and used osmotic pressure data to generate the osmotic v i r i a l c o e f f i c i e n t for correction of nonideal e f f e c t s . More recently, there has been a resurgence of i n t e r e s t i n obtaining a p r a c t i c a l solution to Rinde's equation. Sundelof (52) proposed a method, l a t e r refined by Provencher (41), based on the Fourier convolution theorem. Provencher and Gobush (42) also suggested a combination of quadrature and le a s t squares to solve the Rinde equation, an demonstrated h i s approach by generating synthetic data from a Wesslau d i s t r i b u t i o n , and then using a Schultz function to make his f i t . A l l of the methods that attempted to solve the Rinde equation for the frequency values f(M), suffered from the problem o r i g i n a l l y encountered by Svedberg and Nichols (53); they produced weight fracti o n s that were negative. This problem was overcome independently with t o t a l l y d i f f e r e n t approaches by Donnelly (14,15) and Scholte (47,48). 9. Donnelly (14) recognized that the concentration d i s t r i b u t i o n of polymeric solutes was i n the form of a Laplace transform. He was able to convert the Rinde equation to a Laplace transform and obtain the MWD from i t s inverse. The method was t h e o r e t i c a l l y rigorous, worked well for unimodal d i s t r i b u t i o n s and had the advantage of requiring data from only one rotor speed. However, i t was r e s t r i c t e d to continuous d i s t r i b u t i o n s and was unable to handle multimodal systems (15). Scholte (47) proposed a method whereby the Rinde equation-was transformed to a set of l i n e a r equations. By assuming a series of molecular weights that were exponentially spaced, f(M) became the only unknown to be solved for i n the Rinde equation. Through the use of l i n e a r programming with p o s i t i v e constraints, a f e a s i b l e solution could be obtained. Scholte's method required the use of several rotor speeds to provide 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 for multimodal systems. More recently, Gehatia and Wiff (22,23,60-63) developed a sophisticated mathematical treatment of the data to provide MWDs. They recognized the f a c t that the Rinde equation i s an Improperly Posed Problem (IPP), as did other workers (30), and r e a l i z e d that t h i s problem could be circumvented by applying Tikhonov's r e g u l a r i z a t i o n 10. function. The recognition that the Rinde equation i s an IPP explained the problems encountered i n obtaining a d i s t r i b u t i o n from the Rinde equation. Gehatia and Wiff (21) found that r e g u l a r i z a t i o n alone could not resolve more complex d i s t r i b u t i o n s . They then incorporated the method of l i n e a r programming of Scholte (48) and l a t e r used quadratic programming (63). These techniques allowed, i n theory, the analysis of 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, trimodal systems at one speed were probably the l i m i t that could be analyzed. This very sophisticated technique i s unusable f o r most researchers due to the excessive computer time required. Wan (59) evaluated the methods of Donnelly and Scholte, using dextran as a polydisperse sample. The method of Donnelly was found to be more convenient as the c a l c u l a t i o n s were somewhat simpler and only one run at one speed was required. Later, Adams et a l (1,2,50) showed that nonideal systems could be studied by using the l i g h t s cattering second v i r i a l c o e f f i c i e n t B^ g. Magar (30) reviewed the various methods fo r obtaining MWDs from a s t a t i s t i c i a n ' s point of view. He suggested the use of a l e a s t squares method CL^ norm) to be applied to Scholte's method, as a substitute for l i n e a r programming. This was considered to be a better method since there were s t a t i s t i c a l parameters available to evaluate the f i t of the 11. d i s t r i b u t i o n . A l i m i t a t i o n of the norm, however, was the possible appearance of negative weight f r a c t i o n s , making the method u n r e a l i s t i c . If the norm f a i l e d , the method of steepest descent (31) or the simplex method of Nelder and Mead (37). with posi t i v e constraints were alternatives to'be attempted. Magar also recommended the use of curve f i t t i n g techniques (I^ norm or orthogonal polynomials) f o r smoothing the raw u l t r a c e n t r i f u g a l data. To date no researchers have reported the ap p l i c a t i o n of the recommendations made by Magar. 12. PART I. MOLECULAR WEIGHT DETERMINATIONS THEORY Sedimentation 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 i s an experimental technique t h a t f a c i l i t a t e s study o f macro-molecule b e h a v i o r under the i n f l u e n c e o f a moderate g r a v i t a t i o n a l f o r c e . The b a s i c concept i n v o l v e s a balance between the f o r c e s of sedimentation, due t o c e n t r i f u g a l f o r c e , and d i f f u s i o n , due to the development o f a r e g i o n of lowered p o t e n t i a l . By choosing an a p p r o p r i a t e speed, a r e d i s t r i b u t i o n o f the s o l u t e occurs t h a t tends t o :: • approach e q u i l i b r i u m i f r o t o r speed and temperature are kept c o n s t a n t . Although t r u e e q u i l i b r i u m w i l l never be a t t a i n e d , a f t e r a p e r i o d of time, the changes i n 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 become n e g l i g i b l e when measured by the o p t i c a l system. 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 a s o l u t e a t e q u i l i b r i u m can p r o v i d e a l a r g e amount of i n f o r m a t i o n about the system. 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 a l s o has a d i s t i n c t advantage over other methods o f st u d y i n g the beh a v i o r o f macromolecules i n s o l u t i o n because the method has a t h e o r e t i c a l f o u n d a t i o n on the f i r s t and second laws of thermodynamics. The theory o f 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 thermodynamics t o systems under an e x t e r n a l f o r c e . In the case o f 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 , there a re 13. no k i n e t i c q u a n t i t i e s due to s o l u t e movement, thus a v o i d i n g 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. The thermodynamic approach allows f o r the f o r m u l a t i o n of equations capable of d e s c r i b i n g very complex systems t h a t are p r e s s u r e dependent or p o l y d i s p e r s e i n m o l e c u l a r weight or i n p a r t i a l s p e c i f i c volume. 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 has long been as-s o c i a t e d w i t h 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. The t h e o r y developed over the years has l e d t o equations p r o v i d i n g s o l u t i o n s f o r the g e n e r a l case, i n c l u d i n g the i n f l u e n c e of p r e s s u r e , heterogeneous m o l e c u l a r weights and p a r t i a l s p e c i f i c volumes and of n o n i d e a l i t y . Although the theory e x i s t s f o r complex a n a l y s e s , i n g e n e r a l most 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 weights v i a u l t r a -c e n t r i f u g a t i o n has been done w i t h r e l a t i v e l y simple systems, or w i t h ones which a l l o w assumptions f o r s i m p l i f y i n g the complex treatment. An i l l u s t r a t i o n of the d e r i v a t i o n of the ' b a s i c ' sedimentation e q u i l i b r i u m e q u a t i o n from the sedimentation v e l o c i t y flow equation w i l l be p r e s e n t e d . The equations f o r sedimentation 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 are based on the premise t h a t the t o t a l p o t e n t i a l o f the system i s c o n s t a n t , o r t h a t the t o t a l p o t e n t i a l g r a d i e n t a t any r a d i a l p o s i t i o n between the meniscus and the c e l l bottom has to be zero. S t a r t i n g w i t h the flow equation f o r sedimentation i n the c e n t r i f u g e f o r q s o l u t e s as given by F u j i t a (18), the f o l l o w i n g r e l a t i o n f i e l d : describes the p o t e n t i a l gradient i n a c e n t r i f u g a l Sy „ 4 /Sy \ 6c . — i = f i - v , p ) u ^ r - £ — 1 Eq. 3 6r j=l\6c./ p 6r (k=0,l...,q, m=l,2...,q) where: q - t o t a l number of solutes y^ - chemical p o t e n t i a l per gram of 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) Equation 3 i s based on two commonly made assumptions; the system i s independent of pressure (incompressible), and the solute i s a nonelectrolyte. In practice, these assumptions are generally v a l i d , since sedimentation equilibrium i's commonly attained at r e l a t i v e l y low speeds and the charge e f f e c t s can be n u l l i f i e d due to 'swamping' of the charge by addition of a small amount of supporting e l e c t r o l y t e . Equation 3 b a s i c a l l y describes how the chemical p o t e n t i a l (y) of the system changes with r a d i a l distance (r) as the components are sedimented under a c e n t r i f u g a l force. The f i r s t term on the r i g h t hand side of the equation represents the sedimentation forces acting on the k components. The second term on the r i g h t hand side contains a summation representing the counteracting force of d i f f u s i o n . At sedimentation equilibrium there i s no net transport and no p o t e n t i a l gradient, therefore, the two terms can be equated as follows: 15. - 2 V- d C k Eq. 4 (1 -v ip)u r =2_jV±] - k d r k=l d (i=l,2...,q) where: ^ i k | 5 p i \ Eq. 5 6 c k ' T,P,c m Equation 4 represents q f i r s t order d i f f e r e n t i a l equations that can be solved for q solute concentrations as functions of r a d i a l distance (r) . This cezieralized form of the equation i s rather d i f f i c u l t to solve. By s t a r t i n g with a much simpler two component system (solvent and solute), Equation 4 reduces to: (1 - v l P ) W 2 r = P l ^ 1 E q . 6 dr U t i l i z i n g the assumption that the solute i s a nonelectrolyte, the chemical p o t e n t i a l can be written: , o, ± RT U l = { U l } c + — ln(y c ) Eq. 7 M l where: (u°) - i s the reference p o t e n t i a l per gram of 1 c solute 1 so that the a c t i v i t y c o e f f i c i e n t on the c - concentration scale approaches unity as c^ tends to 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.. S u b s t i t u t i n g Equation 7 i n t o 6 y i e l d s : M ^ ( l - V-^p)w r c RT 1 + c 61ny-6c, T,P J dc. dr Eq. 8 The r i g h t hand s i d e of the e q u a t i o n r e p r e s e n t s the n o n i d e a l c o n t r i b u t i o n s due t o c o n c e n t r a t i o n . I f the c o n c e n t r a t i o n of the s o l u t e i s low enough and the s o l u t e thereby expresses i d e a l b e h a v i o r , t h i s term reduces to u n i t y and can be ignored. The n a t u r a l l o g f u n c t i o n of the a c t i v i t y co-e f f i c i e n t i s a complex f u n c t i o n o f m o l e c u l a r weight and c o n c e n t r a t i o n and can be expressed as an expansion o f a T a y l o r s e r i e s : 2 2 lny = M i B i c i + M l B 2 c l + Eq. 9 S u b s t i t u t i n g t h i s expansion term back i n t o E q u a t i o n 8 produces: M n (1 - v 1 p ) to r c RT 1 + (BjM^ + V 1 ) c 1 + ( 2 B 2 M l B i M i V ^ i ) c 2 + d c dr Eq. 10 T h i s e q u a t i o n p r o v i d e s the b a s i s f o r the thermodynamic a n a l y s i s of sedimentation e q u i l i b r i u m data f o r b i n a r y s o l u t i o n s . I f the c o n c e n t r a t i o n o f the component under 17. study i s s u f f i c i e n t l y high to cause the nonideality term to become s i g n i f i c a n t , evaluation of the nonideality c o e f f i c i e n t , B, i s required. In general, proteins tend to express i d e a l behavior i f correct conditions and low concentrations are used (46). Under these conditions Equation 10 can' be further s i m p l i f i e d to: __i = V 1 - y ^ ^ y E q. i i dr ..RT This equation can then be transformed into the following experimentally useful form: d l n C l = M l ( l - v l P ) co 2 E q > 1 2 d ( r 2 ) 2RT When UV optics are used, the above equation i s d i r e c t l y applicable to the determination of molecular weights. The slope of a p l o t of In absorbance vs. r a d i a l distance squared i s d i r e c t l y proportional to molecular weight. The molecular weight obtained i s only an apparent value since the e f f e c t of concentration has not been taken into account. A range of concentrations could be run to determine the presence of concentration dependence. 18. EXPERIMENTAL U l t r a c e n t r i f u g a t i o n The standard c e n t r i f u g e used f o r a n a l y t i c a l work i s the Beckman Model E u l t r a c e n t r i f u g e (4). T h i s s o p h i s t i c a t e d instrument was not a v a i l a b l e i n our l a b o r a t o r y ; however, an 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 equipped w i t h a newly marketed UV scanning a c c e s s o r y was used. The d e s i g n o f the UV a c c e s s o r y was based on p r i n c i p l e s employed i n the Model E UV Scanner, and p r o v i d e d the L2-65B u n i t w i t h a n a l y t i c a l c a p a b i l i t i e s . Numerous f e a t u r e s a v a i l a b l e on the Model E, such as v a r i a b l e wavelength, s i n g l e s e c t o r scanning and an absorbance range of 0.0 - 2.0 were not p r e s e n t i n t h i s L2-65B v e r s i o n , but i n other r e s p e c t s the scanner was comparable and an economical a l t e r n a t i v e t o i t s more expensive c o u n t e r p a r t . The UV scanning a c c e s s o r y c o n s i s t e d o f t h r e e major components, the i n s i d e chamber o p t i c a l u n i t , the door mounted scanner and the e l e c t r o n i c s u n i t ( m u l t i p l e x e r ) . The remaining equipment ne c e s s a r y t o run the c e n t r i f u g e , such as c e l l s and counterbalances were the same as f o r the Model E. The scanning a c c e s s o r y was a b l e to scan 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 of the s o l u t e i n the c e n t r i f u g e c e l l , thereby measuring the amount of UV r a d i a t i o n 19. a b s o r b e d a t v a r i o u s r a d i a l p o s i t i o n s . The l i g h t s o u r c e was a s h o r t a r c m e r c u r y lamp f r o m w h i c h a n a r r o w band o f w a v e l e n g t h s , a r o u n d 280 nm, was i s o l a t e d b y an i n t e r -f e r e n c e f i l t e r . The l i g h t was d i r e c t e d o n t o t h e s p i n n i n g r o t o r (see s c h e m a t i c o f o p t i c a l s y s t e m , F i g u r e 1) and p a s s e d t h r o u g h t h e c e l l , w i t h t h e image o f t h e c e l l b e i n g f o r m e d a t t h e e n t r a n c e s l i t t o t h e p h o t o m u l t i p l i e r t u b e . The p h o t o m u l t i p l i e r was mounted on a m o b i l e c a r r i a g e d r i v e n by a c o n s t a n t s p e e d m o t o r t h a t was a b l e t o s c a n t h e image. E l e c t r i c a l s i g n a l s r e c e i v e d f r o m t h e sample and r e f e r e n c e s e c t o r s o f t h e c e l l were p a s s e d f r o m t h e p h o t o m u l t i p l i e r t u b e t h r o u g h a m u l t i p l e x e r where t h e s i g n a l s were compared an d t h e d i f f e r e n c e r e c o r d e d on a s t r i p c h a r t r e c o r d e r . The s c a n n i n g u n i t , t h e r e f o r e , w o r ked e s s e n t i a l l y as a d o u b l e beamed s p e c t r o p h o t o m e t e r , w i t h t h e r o t o r a c t i n g as a l i g h t beam c h o p p e r . The m u l t i p l e x i n g u n i t a l s o a l l o w e d s c a n n i n g o f m u l t i -p l a c e d r o t o r s by d e c i p h e r i n g a c o d e d r i n g on t h e b o t t o m o f t h e r o t o r . The s t r i p c h a r t r e c o r d i n g , c a l l e d a ' s c a n ' , g a v e a t r a c e o f t h e 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 t h e c e l l a t e q u i l i b r i u m , a l o n g w i t h a 1.0 a b s o r b a n c e s i g n a l a n d c e l l r e f e r e n c e e d g e s ( F i g u r e 2). The UV s c a n n i n g s y s t e m ha d one m a j o r a d v a n t a g e o v e r t h e o t h e r o p t i c a l s y s t e m s , i n t h a t t h e o u t p u t was i n t h e f o r m o f an e l e c t r i c a l s i g n a l r a t h e r t h a n an o p t i c a l image. T h i s s i g n a l was 20. l i g h t source p h c t a r o i l t i p l i e r tube 0 S l i t BBSS I \ I \ I l i g h t p a t h , s p h e r i c a l m i r r o r rotor Figure 1. Schematic diagram of the UV optical system. H Figure 2. Recorder trace (scan) of a typical seclimentation equilibrium experiment using a double sector ce l l . A - 1.0 absorbance signal B - outer reference edge C - counterbalance darkspace D - solute distribution pattern E - solution - solvent meniscus F - air - solvent meniscus G - air baseline H - counterbalance darkspace I - inner reference edge amenable to d i g i t i z a t i o n and therefore to automated data processing techniques. P r i o r to 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 of the ultracentrifuge was compared to a spectro-photometer for l i n e a r i t y . This was done using a c a r e f u l l y d i l u t e d series of colchicine solutions to produce a standard curve of concentration vs. absorbance. These i n d i v i d u a l d i l u t i o n s were then loaded i n t o the centrifuge, spun at 12,000 rpm and the absorbance recorded. The preparative UV scanner response was found to be lin e a r up to an absorbance of 1.2. The absorbance values were found to d i f f e r by a constant that was e s s e n t i a l l y equivalent to the correspondingly longer path length i n the centrifuge c e l l . A check of the interference f i l t e r showed that it. transmitted only a r e l a t i v e l y narrow band of wavelengths, with a maximum i n t e n s i t y at 2 78 nm. Other checks, such as rotor wobble at low speeds, o i l deposition from the drive, baseline consistency, and r e p r o d u c i b i l i t y of the reference edges, c e l l bottom and meniscus traces, were also made, and shown to be within tolerance. Special attention was paid to the cleanliness of the o p t i c a l system with a l l components outside the housing cleaned a f t e r every run. The o p t i c a l tower was sealed to prevent a d i f f u s i o n of o i l onto the mirrors, so that the unit required disassem bly only after extended periods of time. An addit i o n a l motor was purchased to d r i v e the scanner a t a slower speed, thus a l l o w i n g the data a c q u i s i t i o n system t o r e c o r d more data per scan. The 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 a v a i l a b l e p r o t e i n s of a n a l y t i c a l grade. A complete l i s t can be found i n the Appendix, S e c t i o n A. A l l of these p r o t e i n s were checked f o r p u r i t y by p o l y -a c r y l a m i d e d i s c g e l e l e c t r o p h o r e s i s (13). In a l l cases, the p r o t e i n s were d i a l y s e d f o r a t l e a s t 24 hours a g a i n s t the b u f f e r they were d i s s o l v e d i n , then d i l u t e d t o a p p r o p r i a t e absorbance v a l u e s (280 nm) t h a t were measured u s i n g a Beckman DB spectrophotometer. F o r m o l e c u l a r weight d i s t r i b u t i o n s t u d i e s , the c e l l assembly and l o a d i n g procedures d e s c r i b e d by Chervenka (8) were f o l l o w e d , u s i n g r e g u l a r 12 mm c a r b o n - f i l l e d epoxy r e s i n double s e c t o r c e l l s . F l u o r o c a r b o n s o l u t i o n was added t o these 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 the c e l l bottom. The t h r e e p l a c e Yphantis c e l l was u t i l i z e d f o r simple molecular weight d e t e r m i n a t i o n s and for a s s e s s i n g c o n c e n t r a t i o n dependency. In o r d e r t o a v o i d a leakage problem w i t h t h i s c e l l , a s p e c i a l procedure, d e s c r i b e d i n the Appendix, S e c t i o n B, was f o l l o w e d . Normally, a b l a c k anodized f o u r p l a c e r o t o r (AN-F:> was used which h e l d t h r e e c e l l s and a co u n t e r b a l a n c e . A l l runs were performed a t 20°C w i t h the temperature b e i n g monitored by an i n f r a r e d s e n s i t i v e radiometer l o c a t e d u n d e r t h e r o t o r . The s p e e d a t w h i c h a sample was t o be r u n was e s t i m a t e d f r o m t h e g r a p h o f rpm v s . m o l e c u l a r w e i g h t on p a g e 45 o f C h e r v e n k a ' s M a n u a l o f Methods f o r t h e A n a l y t i c a l C e n t r i f u g e ( 8 ) . The a c t u a l r u n n i n g s p e e d was d e t e r m i n e d by a v e r a g i n g a t l e a s t f i v e r e a d i n g s f r o m t h e odometer. The t i m e t o r e a c h e q u i l i b r i u m was e s t i m a t e d u s i n g t h e c a l c u l a t i o n o f Van H o l d e and B a l d w i n (54) , and a f t e r t h e c a l c u l a t e d t i m e had e l a p s e d , s e v e r a l s c a n s were taker, a t h o u r l y i n t e r v a l s , and compared v i s u a l l y . I f no o b s e r v a b l e c h a n g e s h a d o c c u r r e d , t h e s y s t e m was 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 and a f i n a l s c a n was t a k e n . When a r u n was c o m p l e t e d , t h e r o t o r was a c c e l e r a t e d t o maximum s p e e d t o d e p l e t e t h e c e l l o f a l l p r o t e i n m a t e r i a l . When d e p l e t i o n was c o m p l e t e d , t h e r o t o r was d e c e l e r a t e d t o r u n n i n g s p e e d a n d a f i n a l s c a n was t a k e n f o r u s e as t h e t r u e b a s e l i n e . The d e c e l e r a t i o n p r o c e d u r e was r e q u i r e d t o a v o i d e r r o n e o u s r e s u l t s c a u s e d b y d i s -t o r t i o n o f t h e q u a r t z windows a t h i g h s p e e d s . D a t a P r o c e s s i n g One o f t h e o b j e c t i v e s o f t h i s s t u d y was t o u t i l i z e a r e c e n t l y a c q u i r e d d a t a a c q u i s i t i o n s y s t e m , i n o r d e r t o e l i m i n a t e t h e manual p r o c e d u r e s i n v o l v e d i n o b t a i n i n g d a t a p o i n t s f r o m t h e UV s c a n n e r s t r i p c h a r t o u t p u t . D i g i t i z e d data obtained and stored by the data a c q u i s i t i o n system could then be processed through the use of a desk-top computing system. This system replaced the rather tedious calculations associated with molecular weight determinations with an automated, routine procedure, increasing the accuracy and reducing human error. 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 consisted of three components, a Schlumberger Solatron data transfer unit, a Solatron A220 d i g i t a l voltmeter and a Texas Instruments S i l e n t 700 ASR elec t r o n i c data terminal (teletype) with a cassette tape accessory. A l l three components were mounted on a laboratory c a r t to make the unit self-contained and mobile (Figures 3 & 4). The data transfer unit had a time c o n t r o l l a b l e multichannel c a p a b i l i t y and served to receive and d i g i t i z e the incoming voltage s i g n a l . The d i g i t a l voltmeter displayed the voltage and transferred the data to the teletype. The teletype i n turn printed a hard copy of the d i g i t i z e d values on heat sens i t i v e paper, ,and simul-taneously recorded the data on magnetic tape. The Figure 3. Photograph of the data a c q u i s i t i o n system on a laboratory c a r t - t e l e t y p e , voltmeter and d i g i t i z e r . Figure 4. Photograph of the acquisition system, multiplexing unit and ultracentrif uge. 28. teletype was able to interface d i r e c t l y to a Monroe 1880 programmable desktop c a l c u l a t o r that was used to process the data, i f provided with the appropriate program. A schematic diagram of the data processing operation i s presented i n Figure 5. 2. Desktop computation and p l o t t i n g Most of the ca 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 S c i e n t i f i c P r i n t i n g Calculator. The basic unit came with 512 program steps and 64 main data r e g i s t e r s . For t h i s work, the program step capacity and memory were expanded to the maximum of 4,096 and 512 respectively. The c a l c u l a t o r had a magnetic card reader for preprogrammed instructions or stored data. The p r i n t i n g u n i t allowed a hard copy of the c a l c u l a t i o n r e s u l t s to be obtained and could also l i s t programs that had been written. Most of the s c i e n t i f i c functions were available and a low l e v e l programming language allowed programs to be written on the basis of simple algebraic l o g i c . Symbolic addressing, i n d i r e c t addressing, sub-routines, flagging and decision c a p a b i l i t i e s were avai l a b l e . With these features, the Monroe 1880 c a l -culator provided a very powerful computing f a c i l i t y at a low cost. Centrifuge S t r i p Chart Recorder D i g i t i z e r and Voltmeter Magnetic Tape Teletype Hard Copy Pr i n t e r Monroe 1880 Calculator P l o t t e r Figure 5. Schematic diagram of the data processing operations. 30. A p l o t t i n g u n i t was also obtained to interface with the Monroe 1880. This Monroe PL-2 x-y p l o t t e r could be controlled by the c a l c u l a t o r through a p l o t t i n g program or the i n s e r t i o n of a p l o t t i n g routine into the programs that required the p l o t t i n g of data. The p l o t t i n g c a p a b i l i t y enhanced the usefulness of the c a l c u l a t o r 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 often required to decide on further processing steps. 3. Application of the data a c q u i s i t i o n system to the centrifuge As was mentioned previously, the preparative UV scanner was amenable to the data a c q u i s i t i o n system, since the scanner produced a DC voltage that was d i r e c t l y related to the absorbance difference between the two sectors of the centrifuge c e l l . One problem associated with scanner data obtained by the a c q u i s i t i o n system was that no information was provided on r a d i a l distance, the x coordinate for the c a l c u l a t i o n of molecular weights. This problem was overcome by c a l i b r a t i o n of the constant speed scanning motor movement i n r e l a t i o n to the signal pickup i n t e r v a l of the data transfer unit. In t h i s procedure, the i n i t i a l point i n the scan was i d e n t i f i e d and a constant 31. was applied to convert the time-based data i n t e r v a l s to r a d i a l distance in the centrifuge c e l l . In order to derive s u f f i c i e n t data points per scan, a slower scanning motor was i n s t a l l e d , since the o r i g i n a l motor was designed • for v e l o c i t y u l t r a c e n t r i f u g a t i o n experiments. B. Software A major programming e f f o r t was undertaken to allow for complete processing of the raw data a v a i l a b l e from the centrifuge. Of the many programs written for the c a l -culations 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 outlined here, with the others being b r i e f l y described i n the Appendix, Section C. These three programs are of primary inte r e s t since they i l l u s t r a t e the a b i l i t y of the desktop calculator to automate the c a l c u l a t i o n of molecular weights. 1. Data conversion program This program was d i r e c t l y involved i n c a l c u l a t i n g the natural log of absorbance (In A) and r a d i a l distance 2 squared (x ) terms d i r e c t l y from the data provided by the a c q u i s i t i o n system, u t i l i z i n g the constant derived from 2 the scanning motor. The In A vs. x pairs were stored i n reserved registers to allow other programs to work on t h e t r a n s f o r m e d raw d a t a . 2. A u t o m a t i c l i n e a r r e g r e s s i o n and p l o t t i n g p r o g r a m T h i s p r o g r a m h a d t h e c a p a b i l i t y o f u s i n g d a t a f r o m a d e f i n e d memory r e g i s t e r i n t e r v a l i n a p l o t t i n g - r o u t i n e and a l e a s t - s q u a r e s l i n e a r r e g r e s s i o n p r o c e d u r e . T h i s p r o g r a m was u s e f u l f o r d e t e r m i n i n g t h e m o l e c u l a r w e i g h t o f homogeneous s y s t e m s f r o m t h e c o n s t a n t s l o p e 2 o f t h e l n A v s . x r e l a t i o n s h i p . The p l o t a l l o w e d v i s u a l e x a m i n a t i o n o f t h e l e a s t - s q u a r e s l i n e a r f i t t o t h e d a t a . I f o n l y a p o r t i o n o f t h e d a t a was l i n e a r , a r e g r e s s i o n c o u l d be r e a d i l y r e p e a t e d u s i n g o n l y t h a t p a r t o f t h e s t o r e d d a t a . 3. A u t o m a t i c m u l t i p l e r e g r e s s i o n p r o g r a m w i t h b a c k c a l c u l a t i o n f e a t u r e T h i s p r o g ram was a b l e t o r e t r i e v e s t o r e d d a t a and f i t them by any d e s i r e d p o l y n o m i a l . The r e g r e s s i o n p r o v i d e d t h e p o l y n o m i a l c o e f f i c i e n t s w h i c h , i n t u r n , c o u l d be e n t e r e d i n t o a back c a l c u l a t i o n s u b r o u t i n e t h a t u s e d 2 t h e o r i g i n a l x v a l u e s t o g e n e r a t e t h e b e s t f i t d a t a o r t h e i r d e r i v a t i v e s . T h i s p r o g r a m was u s e f u l f o r t h e e v a l u 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 h e t e r o g e n e o u s s y s t e m s . These three programs i n the Monroe 18 80 calculator could be used to convert the raw data coming from the a c q u i s i t i o n system into molecular weights with a minimum amount of manipulations. Figure 6 i l l u s t r a t e s the data flow i n and out of the programs discussed above. RESULTS As th? UV scanner was a newly marketed accessory, no publications had appeared concerning i t s c a p a b i l i t i e s . Therefore, p r i o r to attempting the more complex investigation of protein mixtures, the preparative UV scanning accessory and the ultracentrifuge had to be tested for accuracy i n evaluating molecular weights. Homogeneous Systems Experiments were performed to determine the molecular weights of various standard proteins and to check for the presence of protein concentration dependency. These i n i t i a l experiments were also used to te s t and develop the data a c q u i s i t i o n system, with the ca l c u l a t i o n s performed both manually and by the automated system, i n order to evaluate the r e l a t i v e accuracy of the two approaches. In these experiments, a l l of the proteins were run at a wide range of concentrations (usually a s e r i a l d i l u t i o n ) to Linear Regression and Plotting.. Slope (derivative) Molecular Weight Raw Data Data Conversion Program Stored „ In A vs. x Back Calculation Smooth In A vs. x" M u l t i p l e Regression Program Coefficients Derivatives Molecular Weights Figure 6. Schematic diagram of data flow through the data conversion, linear regression and multiple regression programs. d e t e r m i n e w h e t h e r t h e m o l e c u l a r w e i g h t was c o n c e n t r a t i o n d e p e n d e n t . A f t e r a s s e s s i n g t h e p u r i t y b y p o l y a c r y l a m i d e e l e c t r o p h o r e s i s , f o u r homogeneous p r o t e i n s , o v a l b u m i n , c o n a l b u m i n , l y s o z y m e and t r y p s i n i n h i b i t o r were c h o s e n f o r 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 . The r e s u l t s o b t a i n e d f r o m o v a l b u m i n w i l l s e r v e t o i l l u s t r a t e t h e c a p a b i l i t i e s o f t h e UV s c a n n i n g s y s t e m when l i n k e d t o t h e d a t a p r o c e s s i n g s y s t e m . N i n e d i f f e r e n t c o n c e n t r a t i o n s o f o v a l b u m i n i n 0.05M, pH p h o s p h a t e b u f f e r , r a n g i n g i n c o n c e n t r a t i o n f r o m 0.20 t o 0.60 i n a b s o r b a n c e , were l o a d e d i n t o t h r e e Y p h a n t i s c e l l s , a n d r u n i n a m u l t i p l a c e r o t o r a t a p p r o x i m a t e l y 12,000 rpm. The t i m e t o r e a c h e q u i l i b r i u m was c a l c u l a t e d u s i n g t h e programmed v e r s i o n o f t h e Van H o l d e - B a l d w i n e q u a t i o n (54) (see A p p e n d i x , S e c t i o n C ) . Upon r e a c h i n g e q u i l i b r i u m , a l l c e l l s were s c a n n e d i n s u c c e s s i o n , w i t h t h e d a t a b e i n g d i g i t i z e d and p u t on t a p e by t h e d a t a a c q u i s i t i o n s y s t e m . An example o f t h e h a r d c o p y p r o d u c e d f o r e a c h s c a n c a n be s e e n i n F i g u r e 7. The n o r m a l p r o c e d u r e f o r t h e c a l c u l a t i o n o f m o l e c u l a r w e i g h t s was t o r e a d t h e a b s o r b a n c e v a l u e s f r o m t h e s t r i p c h a r t , r e c o r d t h e g r i d p o s i t i o n and c o n v e r t t h e s e v a l u e s t o l n a b s o r b a n c e and r a d i a l d i s t a n c e s q u a r e d . Once t h e s e d a t a had b e e n a c c u m u l a t e d , t h e y were p l o t t e d and e n t e r e d i n t o a l i n e a r r e g r e s s i o n p r o g r a m . I f t h e p l o t was l i n e a r , w h i c h i t s h o u l d be i f t h e s o l u t e was p s e u d o -DATE - - - 12/12/76 RUN# 36 SAMPLE - - - - - - - - - OVALBUMIN SCAN# 3 O.D. _ _ _ _ _ _ _ _ _ 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 OF SCAN #3 F i g u r e 7. F a c s i m i l e o f t h e t e l e t y p e o u t p u t . 37 . 2 i d e a l ( 2 0 ) , t h e s l o p e o f t h e p l o t ( d i n A / d ( x )) was t h e n d i r e c t l y p r o p o r t i o n a l t o m o l e c u l a r w e i g h t ( E q u a t i o n 1 2 ) . The a b o v e p r o c e d u r e s , i f c a r r i e d o u t m a n u a l l y f o r s e v e r a l s a m p l e s became v e r y t e d i o u s and s u b j e c t t o e r r o r . U s i n g t h e a c q u i s i t i o n s y s t e m , a l l p r o c e d u r e s c o u l d b e c a r r i e d o u t a u t o m a t i c a l l y . A f t e r t h e s c a n s o f t h e s a m p l e s a t e q u i l i b r i u m were c o m p l e t e d , t h e t e l e t y p e was i n t e r f a c e d w i t h t h e Monroe 1880 c a l c u l a t o r and t h e d a t a were t r a n s f e r r e d i n t o t h r d a t a r e g i s t e r s o f t h e c a l c u l a t o r by a c o m m u n i c a t i o n s p r o g r a m . 2 The c o n t r o l was t h e n p a s s e d t o t h e l n A v s . x p r o g r a m t h a t u s e d some b a s i c i n p u t o b t a i n e d f r o m t h e s c a n t o p r o d u c e a s e t o f v a l u e s o f l n a b s o r b a n c e v s . r a d i a l d i s t a n c e s q u a r e d ( F i g u r e 8 ) . T h i s o u t p u t , w h i c h was n o t n o r m a l l y p r i n t e d , was s t o r e d i n a r e s e r v e d s e t o f d a t a r e g i s t e r s t h a t c o u l d be a c c e s s e d by a p l o t t i n g r o u t i n e , w h i c h i n t u r n p l o t t e d t h e d a t a s e t ( F i g u r e 9 ) . C o n c u r r e n t l y , t h e same d a t a were u s e d by a r e g r e s s i o n p r o g r a m t h a t computed t h e s l o p e and o t h e r s t a t i s t i c a l p a r a m e t e r s ( F i g u r e 8 ) . A f t e r o b t a i n i n g t h e s l o p e , c o n t r o l was p a s s e d t o a s m a l l s u b -r o u t i n e t h a t c a l c u l a t e d t h e m o l e c u l a r w e i g h t , a f t e r t h e a n g u l a r v e l o c i t y and p a r t i a l s p e c i f i c v o lume were s u p p l i e d . A l l o f t h e p r o g r a m s m e n t i o n e d , t h e c o m m u n i c a t i o n s p r o g r a m , 2 I n A v s . x , l i n e a r r e g r e s s i o n w i t h p l o t t i n g and t h e m o l e c u l a r w e i g h t c a l c u l a t i o n s u b r o u t i n e , were p r e s e n t i n Figure 8. Input and output of the data conversion program and l i n e a r regression plus p l o t t i n g program. Data conversion program Input: A - number of data points - f i r s t good absorbance value - baseline correction - baseline - p o s i t i o n of f i r s t good value 2 1 Output: B - x and l n A pairs (not normally printed) Linear regression plus p l o t t i n g program 2 Input: C - maximum and minimum for x - maximum and minimum for l n A - number of data points to be regressed - location of f i r s t data pair i n memory 2 Output: D - mean of x - mean of l n A - slope - intercept - standard deviation - c o r r e l a t i o n c o e f f i c i e n t Calculation of molecular weight Input: E - (1 - v p ) - X x 10 6 Output: F - molecular weight Ln A values represent the natural log of the voltage produced by the scanner, rather than true absorbance. 3 9 . 2 3 « 0 0 0 0 0 0 0 17 7- 0 0 0 0 0 0 0 I 3 « - 0 0 0 0 0 0 0 A 4 2 < . o o o o o o o I 7 . 1 5 5 8 0 0 0 J 5 0 . 9 3 3 6 1 5 3 B — ' 3 • 0 7 2 6 9 3 3 5 0 • 7 7 2 2 1 7 2 3 . 0 2 5 2 9 1 0 5 0 • 5 5 6 2 7 9 3 2 • 9 1 2 3 5 0 6 5 0 • 3 4 0 8 0 1 6 2 . 3 1 5 4 0 8 7 5 0 • 1 2 5 7 8 4 0 2 • 7 5 3 6 6 0 7 4 9 • 9 1 12 2 6 6 2 . 6 3 9 0 5 7 3 4 9 . 6 9 7 1 2 9 4 2 -« 5 9 5 2 5 4 7 4 9 < - 4 8 3 4 9 2 4 2 - 4 8 4 9 0 6 6 4 9 • 2 7 0 3 15 5 2 . 4 15 9 13 7 4 9 « 0 5 7 5 9 8 9 . - I 3 3 2 14 3 8 4 S ' 8 4 5 3 4 2 4 2 « 2 5 12 9 17 6 3 3 5 4 6 1 2 • 1 6 3 3 2 3 0 4 6 • 4 2 2 2 0 9 9 2 « 1 16 2 5 5 5 4 S « 2 1 1 3 3 4 0 2 • 0 1 4 9 0 3 0 . 4 3 . 0 0 0 9 18 2 t • 9 3 15 2 14 ! 4 7 • 7 9 0 9 6 2 6 1 l • 8 2 4 5 4 9 2 i j 4 7 • 5 8 1 4 6 7 2 i t 1 • 7 4 0 4 6 6 1 ! 1^  5 1 . 0 0 0 0 0 0 0 4 7 - 0 0 0 0 0 0 0 3 • 1 0 0 0 0 0 0 1 . 7 0 0 0 0 0 0 • ••••••••• 1 7 . 0 0 0 0 0 0 0 9 2 * 0 0 0 0 0 0 0 4 9 « 1 6 8 7 8 9 1 A 2 . 3 7 6 0 1 8 6 A 0 • 3 8 9 8 3 7 3 A 1 6 * 7 9 1 8 0 9 4 A 0 - 0 1 5 2 9 5 9 A 0 • 9 9 9 2 0 0 4 A 0 * 2 5 1 0 0 0 0 1 . 5 7 7 8 2 1 0 F - 4-7,9. 9 6 * 1 8 5 9 3 40. the c a l c u l a t o r and the operator passed c o n t r o l and entered the i n p u t f o r each scan. Using t h i s system, t h e c e n t r i f u g a t i o n data f o r a l l n i n e c o n c e n t r a t i o n s o f ovalbumin were processed i n about one hour, making a p r e v i o u s l y long and complicated c a l c u l a t i o n simple and l e s s s u s c e p t i b l e to e r r o r s . The p l o t i l l u s t r a t e d i n F i g u r e 9 was t y p i c a l of the ones o b t a i n e d f o r the ovalbumin run. The molecular weights c a l c u l a t e d are presented i n Table I. These r e s u l t s showed no c o n c e n t r a t i o n dependency under the e x i s t i n g c o n d i t i o n s , and the average m o l e c u l a r weight of 45,300 d a l t o n s compared w e l l w i t h the v a l u e s i n the l i t e r a t u r e ( 4 9 ) . Runs w i t h lysozyme, t r y p s i n i n h i b i t o r and conalbumin produced s i m i l a r r e s u l t s , w i t h t h e i r r e s p e c t i v e m o l e c u l a r weights b e i n g 1 4 , 4 0 0 , 16,300 and 78,000 d a l t o n s . Heterogeneous Systems Pse u d o - i d e a l heterogeneous systems c o u l d a l s o be handled r o u t i n e l y by the data p r o c e s s i n g system. In the case of heterogeneous systems, the m o l e c u l a r weight v a r i e d along the s o l u t i o n column of the c e n t r i f u g e c e l l and a non-2 l i n e a r f u n c t i o n was obtained f o r the l n A v s . x p l o t . In o r d e r t o evaluate the m o l e c u l a r weights o f these systems, 2 the d e r i v a t i v e , d i n A/d(x ), had t o be e v a l u a t e d . 2.9 3 51 radial distance squared (x ) • J-Figure 9. Plot of ln A vs.x 2 for ovalbumin showing the least squares f i t to the data. Table I. Molecular Weights Obtained f o r Ovalbumin* Absorbance Molecular Weight (daltons) 0.20 45,959 0. 25 44,474 0.30 47,592 0. 35 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 molecular weight - 45,300 Standard deviation - - 1,200 P a r t i a l s p e c i f i c volumes were obtained from the Handbook of Biochemistry ( 4 9 ) . This c a p a b i l i t y was provided by the modification of a multiple regression program that allowed automatic f i t t i n g 2 of l n A vs. x data to any desired polynomial. From the derived polynomial c o e f f i c i e n t s , the best f i t function and i t s derivatives could be generated and plotted. The s a l i e n t features 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 using experimental data obtained from running catalase, an enzyme present i n solution i n both i t s monomer and tetramer forms. Catalase was dissolved i n 0.05 M phosphate buffer, pH and centrifuged at 8,300 rpm u n t i l equilibrium was reached. The centrifuge data were i n i t i a l l y processed as described 2 for homogeneous systems. Since the p l o t of l n A vs. x data showed marked concavity to the x axis, the stored values were analysed using a modified multiple regression program that allowed automatic f i t t i n g of l n A vs. x data by any desired polynomial. In general, a quadratic function was suitable. Figure 10 i l l u s t r a t e s the p l o t of 2 the l n A vs. x data (ci r c l e s ) and the l e a s t squares quadratic f i t ( s o l i d line) obtained for catalase. Figures 11 and 12 i l l u s t r a t e the input and output from the multiple regression c a l c u l a t i o n . The derivative of the function can be r e a d i l y obtained from the back c a l c u l a t i o n feature of the program, using the regression c o e f f i c i e n t s from the r e l a t i o n : Figure 11. Input and output of the multiple regression program. Input: A - x and l n pairs to be regressed B - number of data pairs to be regressed C - number of transforms D - transform order Output: E - means F - standard deviations G - c o r r e l a t i o n c o e f f i c i e n t H - scandaru error of estimate I - F - r a t i o J - f i r s t c o e f f i c i e n t - standard deviation of the c o e f f i c i e n t - t - t e s t 46. 1_ 5 1 4 7 7 3 5 0 0 0 0 0 5 0 3 « 0 0 0 0 0 0 0 0 — C E L 4 9 * 4 8 5 7 1 4 2 6 2,4 4 9 • 9 7 7 5 8 5 1 2 1,3 5 1 • 3 9 9 8 5 . 0 0 5 7 1 4 2 8 1 2 3 - - D 2 . 8 5 8 0 7 3 7 2 0 • 2,7 9 5 * 1 7 2 3 0 6 1,0 7 7 . 4 4 0 8 1 3 M / 1 5 • 0 0 ! 1 0 0* 5 1 * 0 0 2 2 r 1 • 1 0 8 8 2 2 7 5 SO o t 6 » 2 2 F 1 0 9 * 7 6 2 9 5 6 5 so . 0 2 5 0 • 9 5 j L 8,1 5 2 • U 0 6 5 6 SD 0 J 5 • 8 6 i 0 * 5 9 5 3 3 3 3 7 SO 0 4 5 0 • 6 8 5 • 6 3 5 0 • 4 1 5 • 3 9. G — 0 * 9 9 7 8 1 6 3 0 DO. 5 0 • 1 4 H -- 0 * 0 3 1 7 2 2 5 8 e 5 « 2 4 4 9 • 8 8 3 • df o 5 • 0 8 1 0 . df / 4 9 • 6 1 4 • 9 9 I "~ .1,5 2 3 • 1 2 9 5 7 0 F 4 9* 3 5 4 * 7 9 4 9 • 0 8 1 • 4 • 6 9 -4 8 • 8 2 ; r 1 7 4 * 9 3 5 7 3 6 2 A 4 » 5 9 J i 6 5 . 3 3 7 7 2 8 6 2 s 4 8 • 5 6 L 2 * 6 7 7 4 0 7 6 7 t 4 » 4 9 4 8 • 3 0 4 • 4 3 ; - 3 * 6 4 9 3 6 0 0 6 A . 4 3. 0 3 1 . 3 2 0 4 1 2 8 5 S : 4 • 3 8 - 2 * 7 6 3 8 1 7 4 3 t; 4 7 • 7 7 4 • 3 0 0 * 0 2 5 4 1 5 8 1 A 0 0 G 0 0 0 - B 0 . 0 0 8 8 9 2 6 3 s Figure 12. Input and output of the multiple regression back c a l c u l a t i o n . Normal back c a l c u l a t i o n Input: A - number of data pairs - number of transforms - transform order B - c o e f f i c i e n t s derived from the best f i t 2 Output: C - best f i t x and l n A values Derivative c a l c u l a t i o n - as above except that the c o e f f i c i e n t s were changed 1 4 * 0 0 0 0 0 0 0 0 3 . 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 1 8 12 4 4 0 3 9 5 0 0 0 0 0 0 8 9 4 9 8 4 5 7 6 8 0 0 0 0 0 0 6 4 3 0 5 0 3 1 4 1 0 0 0 0 0 0 4 2 2 4 3 9 6 6 1 4 0 0 0 0 0 0 2 3 0 1 5 10 9 8 8 0 0 0 0 0 0 0 6 8 9 5 0 5 2 6 1 0 0 0 0 0 0 9 2 3 5 2 8 3 2 3 5 0 0 0 0 0 0 3 0 18 8 9 6 9 0 8 0 0 0 0 0 0 6 9 17 6 8 19 1 4 2 1 . o o o c o o o o 2 . 0 0 0 0 0 0 0 0 ^ 3 . 0 0 0 0 0 0 0 0 _J •2,7 9 5 • 1 7 2 3 0 6 ~~j 1 7 4 . 9 3 5 7 3 6 2 I 6 4 9 3 8 0 0 6 B 0 2 5 4 1 5 8 1 _ J 1 7 -7 0 5 1 1 5 0 0 5 0 0 5 0 9 5 0 0 4 9 0 4 9 0 4 9 0 4 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 9 3 5 7 3 6 2 2 9 8 7 6 0 1 2 0 7 6 2 4 7 4 3 2 2 0 0 0 0 0 0 1 2 7 4 9 0 9 8 9 5 0 0 0 0 0 0 9 9 4 8 0 2 2 4 6 8 0 0 0 0 0 0 8 7 3 2 3 0 3 7 4 1 0 0 0 0 0 0 7 6 2 7 7 5 3 7 14 0 0 0 0 0 0 6 6 3 4 3 7 2 5 8 8 0 0 0 0 0 0 5 7-8 2 8 5 2 1 6 1 0 0 0 0 0 0 5 0 0 7 6 9 1 1 3 5 0 0 0 0 0 0 4 3 6 6 3 0 8 6 0 8 0 0 0 0 0 0 3 8 0 9 3 6 7 7 y = a + bx + cx + dx Eq. 13 y 1 = b + 2cx + 3dx^ Eq. 14 Therefore, re-entry of the c o e f f i c i e n t s according to Equation 14 produces the deri v a t i v e , and hence the point molecular weights of the solute at equilibrium. The molecular weights obtained i n the above manner are presented i n Table I I . The molecular weights i n the region of the meniscus and those at the c e l l bottom agreed well with the molecular weights of 57,500 and 232,000 daltons for the monomer and tetramer re s p e c t i v e l y (49). DISCUSSION As a method for the determination of molecular weights of proteins, the preparative UV scanner had a d i s t i n c t advantage over other o p t i c a l systems, as i t allowed very low concentrations to be analyzed. The low concentrations, i n general, obviated the problems associated with nonideality, and t h i s was found to be the case i n a l l the proteins examined. A drawback to the o p t i c a l system, however, was i t s s e n s i t i v i t y to 50. T a b l e I I . M o l e c u l a r W e i g h t s O b t a i n e d f o r C a t a l a s e * x (cm ) M o l e c u l a r W e i g h t ( d a l t o n s ) 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 48. 82 80,690 48.56 72,831 48.30 67,456 48.03 64,494 47.77 64,136 * - @ 8,300 rpm - v = .730 UV absorbing contaminants. The correct determination of the baseline a f t e r depletion of the sample at the end of a run was e s s e n t i a l for obtaining reasonable molecular weights. Schachman and Edelstein (46) noted that errors i n the baseline determination could be caused by the presence of UV absorbing low molecular weight components derived from d i a l y s i s tubing. Optical systems using r e f r a c t i v e index (schlieren and raleigh) are not s u f f i c i e n t l y s e n s i t i v e to s i m i l a r contaminants to pick up t h i s contribution. Another problem was that some depletion traces had absorbance values below that of the a i r - b u f f e r baseline. The reason for t h i s was unknown and any runs with t h i s anomaly were discarded. Recently, Beckman Instruments published a technical b u l l e t i n (7) that compared the preparative UV scanner to the Model E scanner. Many of the factors that could a f f e c t the accuracy of the scanner r e s u l t s , such as rotor precesion, o p t i c a l l i n e a r i t y , scanner r e p r o d u c i b i l i t y and performance of the e l e c t r o n i c s of the unit, were investigated. In the f i n a l analysis, the preparative UV scanner was deemed for a l l p r a c t i c a l purposes to be equivalent to the Model E scanner, i f the drive and optics were shown to be up to standard. A further note was made concerning the problem of obtaining the zero baseline value by depletion, the uncertainties of which could lead to v a r i a t i o n s i n m o l e c u l a r w e i g h t e s t i m a t i o n s . As C h e r v e n k a (7) n o t e d , t h i s p r o b l e m was n o t r e s t r i c t e d t o t h e p r e p a r a t i o n UV s c a n n e r , b u t was p r e v a l e n t i n t h e M o d e l E s c a n n e r a l s o . I n g e n e r a l , t h e c o n c l u s i o n s i n t h e Beckman b u l l e t i n c o n c u r r e d w i t h t h e r e s u l t s o b t a i n e d i n t h i s work. The d a t a a c q u i s i t i o n s y s t e m was p r o v e n t o f u n c t i o n v e r y w e l l , r e p r o d u c i n g m a n u a l l y s e l e c t e d d a t a r e a d i l y . The p r o c e s s i n g o f d a t a i n t o m o l e c u l a r w e i g h t s , f o r homogeneous and h e t e r o g e n e o u s s y s t e m s , became a r o u t i n e p r o c e d u r e , r e q u i r i n g o n l y t h e l o a d i n g o f t h e a p p r o p r i a t e p r o g r a m s , t h e p r o v i s i o n o f some b a s i c i n p u t p a r a m e t e r s and t h e p a s s i n g o f c o n t r o l f r o m one p r o g r a m t o a n o t h e r . A l t h o u g h a programming e f f o r t was r e q u i r e d i n i t i a l l y , t h e t i m e c o n s u m i n g and e r r o r p r o n e c a l c u l a t i o n s were e a s i l y p e r f o r m e d o n c e t h e p r o c e d u r e was e s t a b l i s h e d . PART I I . MOLECULAR WEIGHT DISTRIBUTIONS THEORY As mentioned p r e v i o u s l y , sedimentation 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 has the p o t e n t i a l f o r p r o v i d i n g a the system. E a r l y i n the development of the u l t r a -centrifuge,- Rinde (44) d e r i v e d an i n t e g r a l r e l a t i o n t h a t d e s c r i b e d the complete molecular weight d i s t r i b u t i o n o f s o l u t e s when the system was a t e q u i l i b r i u m . The Rinde c e l l s by s t a r t i n g w i t h the molecular weight r e l a t i o n (Equation 12) obtained i n P a r t I. The f o l l o w i n g e q u a t i o n s p e r t a i n o n l y t o i d e a l systems t h a t are i n c o m p r e s s i b l e and have the same p a r t i a l s p e c i f i c volume and e x t i n c t i o n c o e f f i c i e n t . For a multicomponent system, w i t h i n the s t a t e d l i m i t a t i o n s , Equation 12 can be expressed as f o l l o w s : l a r g e amount o f i n f o r m a t i o n about the s o l u t e s c o m p r i s i n g e q u a t i o n can be d e r i v e d f o r s e c t o r shaped c e n t r i f u g e d i n c . 2 Eq. 15 2RT or dc. M. (1 - v . p) oa c • Eq. 16 2RT S i n c e p a r t i a l s p e c i f i c v o l u m e s and e x t i n c t i o n c o e f f i c i e n t s a r e assumed t o be t h e same f o r a i l components, a new t e r m c a n be d e f i n e d : ( 1 " ^ p ) f a ) 2 E q . 17 2RT S o l v i n g E q u a t i o n 16 f o r i components r e s u l t s i n : dc ^ = AcM = A / ^ c i M i E q . 18 d ( r ) 1 1 The f o l l o w i n g d e f i n i t i o n s were i n i t i a t e d by F u j i t a ( 1 9 ) : r, - r - r m Eq . 19 where: b - i s t h e c e l l b o t t o m m - i s t h e s o l u t i o n m e n i s c u s X = (1 - vp)co2(r? - r 2)/2RT t> m Eq. 2 0 Using Equations 19 and 20, Equation 16 can be converted to din c. — = -AM. Eq. 21 dC Integrating 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 i(C)/c I ( 5=0) = -XM.5 l Eq. 22 or: c i(?) = c±U=6) exp(-XM I 5) Eq. 23 Equation 22 has been derived without any reference to the shape of the c e l l or the i n i t i a l concentration of the sampl By using the law of the conservation of mass, the i n i t i a l concentration of each component of the sample of a sector-shaped c e l l can be described. Here 6 i s the sector angle and h the length of the solution column.-56. r b Gh / c , d ( r 2 ) = 6 h c o i ( r m ~ r b } E q " 2 4 o r d£ = c . E q . 25 o i 0 Where C q ^ i s t h e i n i t i a l c o n c e n t r a t i o n o f t h e i component. The t o t a l c o n c e n t r a t i o n o f t h e c e l l c a n be e x p r e s s e d b y summing a l l t h e i n i t i a l c o n c e n t r a t i o n s o f e a c h o f q components: q i cd£ E q . 26 'o ^ ° o i I i = l 0 By s u b s t i t u t i n g E q u a t i o n 23 i n t o E q u a t i o n 25, t h e f o l l o w i n g c a n be o b t a i n e d : .XM.C . „ ' on c.(5=0) = E q ' 2 7 1 - exp(-XM i) S u b s t i t u t i n g E q u a t i o n 27 i n t o E q u a t i o n 23 g i v e s : M.c o. e x p ( - X M . e ) E q _ 2 g c, (V = 1 - exp(-XM i) Summing a l l the components i n the system and r e l a t i n g the t o t a l to the i n i t i a l concentration produces: ( 5) = f i f i _ V > AM. exp(-XM.g) c o . E g > 2 9 1 - exp(-AM o "', ,* .) c 1=1 ^ 1 o Substituting f i = c o i / c for the weight f r a c t i o n of component i , the equation becomes: c(£) X M ± e x p ( - A M I ^ ) c 1 - exp ( - A M . ) o • . , ^ 1 i=l f E c I - 30 i I f the molecular weights of the components of the system represent a continuous d i s t r i b u t i o n , then Equation 30 can be written i n the general form as follows: CO c(5) / AM exp(-AMO = I f(M)dM J J-C Q I 1 - exp (-AM) 0 where f(M) represents a d i f f e r e n t i a l molecular weight d i s t r i b u t i o n of the sample components. This i s the Rinde equation upon which a l l the methods for attempting 58. to o b t a i n a complete MWD have been based and which i s r e s t r i c t e d i n i t s a p p l i c a t i o n t o p s e u d o - i d e a l systems. The Rinde e q u a t i o n has been extended by Soucek and Adams (50) t o i n c l u d e some n o n i d e a l systems by u s i n g the l i g h t s c a t t e r i n g v i r i a l c o e f f i c i e n t (B, ) to c o r r e c t f o r non-^ Is i d e a l i t y . Wan (59) a l s o d e r i v e d equations t h a t allowed the use o f Yphantis c e l l s i n o b t a i n i n g MWDs. Although the Rinde equation was r e c o g n i z e d t o be a u s e f u l development s h o r t l y a f t e r i t s p u b l i c a t i o n i n 1928, a p r a c t i c a l s o l u t i o n d i d not appear u n t i l r e c e n t l y , when two s o l u t i o n s appeared almost s i m u l t a n e o u s l y , one by Donnelly and the other by S c h o l t e (14,47). Both s o l u t i o n s to the Rinde equation circumvented the problem o f n e g a t i v e f r e q u e n c i e s t h a t had appeared i n a l l p r e v i o u s s o l u t i o n s . However, the s o l u t i o n submitted by S c h o l t e proved t o be much more v e r s a t i l e , as i t was capable o f a n a l y z i n g multimodal d i s t r i b u t i o n s . 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 S c h o l t e ' s s o l u t i o n i n v o l v e d a s i m i l a r approach t o some of the e a r l i e r attempts i n i t s use o f simultaneous e q u a t i o n s , b ut d i f f e r e d i n i t s i n t e r p r e t a t i o n o f how to s o l v e the problem. S c h o l t e f i r s t o f a l l expressed Equation 31 as f o l l o w s : 59. q c o i=l Eq. 32 where the subscript j i n f e r s that X i s a v a r i a b l e . Equation 3 2 describes the concentration at any r a d i a l p o s i t i o n t, as the sum of a l l the molecular weights present i n the system at that r a d i a l p o s i t i o n . The concentration at that point c(£) i s related to tl\i frequency of the presence of a l l the contributing molecular weights. As the equation stands there are two unknowns, the molecular weights contributing to the concentration at any desired r a d i a l p o s i t i o n and the frequency of t h e i r presence. A l l the other parameters can be obtained from experimental data. Scholte reduced the two unknowns to one by a r b i t r a r i l y assigning an exponentially spaced molecular weight series covering the whole spectrum of molecular weights present i n the sample, and solved the equation for the frequency values by l i n e a r programming. A detailed: summary of the c a l c u l a t i o n procedure follows: Scholte f i r s t of a l l rewrote Equation 32 as; f. I Eq. 33 o 60. where X .M. exp (-X .M. E,) K = -2-Ji ___ E q > 34 1 3 1 - exp(-AjM i) 6j i s a slack variable included to allow for experimental error. By expanding Equation 33, a set of simultaneous equations i s formed that accounts for the changing concentration i n r e l a t i o n to r a d i a l distance. C ^ l ) / c o = V l l + f2 K21 + f3 K31 + + 61 c ( C 2 ) / c Q = f x K 1 2 + f 2 K 2 2 +. f 3 K 3 2 + •-• + 6 2 Eq. 35 c(E )/c = f,K, + f K- + f-K- + ••• + 6. n o 1 l n 2 2n 3 3n j A measurement of the concentration for a given lambda (a function of rotor speed) at any r a d i a l p o s i t i o n would provide one c(£)/c value. Therefore, the analysis of u l t r a c e n t r i f u g a l data at various speeds, and the measure-ment of the concentration at a number of r a d i a l positions provide a l l the data required except for the molecular weights. The provision of a molecular weight series would allow the c a l c u l a t i o n of the K^j term i n each equation, l e a v i n g t h e f r e q u e n c y v a l u e as t h e o n l y unknown. The s e t o f o v e r d e t e r m i n e d s i m u l t a n e o u s e q u a t i o n s i l l u s t r a t e d b y E q u a t i o n 3 5 c o u l d t h e n b e s o l v e d b y l i n e a r p r ogramming. To a v o i d n e g a t i v e f r e q u e n c i e s , S c h o l t e p r o p o s e d t h e u s e t o be o b t a i n e d a n d a t t h e same t i m e m i n i m i z e d t h e e r r o r t e r m 6.. The f i n a l s o l u t i o n w o u l d e x p r e s s t h e f r e q u e n c y 3 o f e a c h o f t h e m o l e c u l a r w e i g h t s a r b i t r a r i l y c h o s e n t o t h e c o m p l e t e r a n g e p r e s e n t i n t h e s y s t e m , t h e sum o f t h e f r e q u e n c i e s o f t h e s o l u t i o n w o u l d e q u a l o ne. The s o l u t i o n o b t a i n e d i n t h i s manner was n o t u n i q u e . S c h o l t e u s e d t h i s p r o p e r t y o f t h e c a l c u l a t i o n t o a d v a n t a g e b y s o l v i n g t h e same s e t o f s i m u l t a n e o u s e q u a t i o n s a g a i n , b u t u s i n g a n o t h e r s e t o f m o l e c u l a r w e i g h t s t h a t were s h i f t e d o v e r b y 1/4 . , a f a c t o r o f 2 ' f r o m t h e i n i t i a l s e r i e s . T h i s p r o c e d u r e was r e p e a t e d a t o t a l o f f o u r t i m e s , p r o v i d i n g a l a r g e r s e t o f s o l u t i o n s t h a t c o u l d b e p l o t t e d a g a i n s t m o l e c u l a r w e i g h t t o p r o v i d e a c o m p l e t e 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 . S t a t e d m a t h e m a t i c a l l y : o f p o s i t i v e c o n s t r a i n t s . T h i s a l l o w e d a r e a l i s t i c s o l u t i o n r e p r e s e n t t h e s y s t e m . I f t h e m o l e c u l a r w e i g h t s c o v e r e d E f i = 1 f o r any one s e r i e s E q . 36 4 f o r a l l s e r i e s E q . 37 therefore: X)f.j/4 = 1 for a l l series Eq. 38 Through experience Scholte found that an exponential molecular weight series with a spacing of multiples of two between molecular weights provided the best sol u t i o n . In summation, Scholte's procedure for obtaining the complete MWD involved: 1. A r b i t r a r i l y s electing a molecular weight series covering the range of the molecular weights present i n the sample. 2. Calculating the K^j terms using the molecular weight series and equating them to the C ( ^ ) / C q values. 3. Solving the simultaneous equations by l i n e a r programming with p o s i t i v e constraints. 4. Increasing the o r i g i n a l molecular weight series by 2 1' 4. 5. Repeating steps 2-4 three more times. 6 . P l o t t i n g the frequencies obtained against molecular weight to produce a complete MWD. Multiple Regression as an Alternative Solution Magar (30) reviewed a l l the major methods of obtaining complete MWDs v i a equilibrium u l t r a c e n t r i f u g a t i o n . With reference to the method of Scholte, Magar suggested the use o f m u l t i p l e r e g r e s s i o n as an a l t e r n a t i v e t o l i n e a r programming f o r o b t a i n i n g complete MWDs. The major argument i n f a v o r of m u l t i p l e r e g r e s s i o n was the a v a i l a b i l i t y o f s t a t i s t i c a l parameters such as standard e r r o r s , c o n f i d e n c e 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 t h a t c o u l d h e l p e s t a b l i s h the f i t o f the d i s t r i b u t i o n t o the o r i g i n a l d a t a . T h i s would enable the r e s e a r c h e r t o s t a t e the degree of p r o b a b i l i t y w i t h which the d i s t r i b u t i o n f u n c t i o n approximated the t r u e d i s t r i b u t i o n . Although these c o n s i d e r a t i o n s f o r the use of m u l t i p l e r e g r e s s i o n were reasonable, t h e r e i s one major d e t e r r e n t . T h i s i s the appearance o f ne g a t i v e f r e q u e n c i e s i n the s o l u t i o n , a problem avoided i n the l i n e a r programming method o f Sc h o l t e . Since p o s i t i v e c o n s t r a i n t s c o u l d not be b u i l t i n t o l i n e a r r e g r e s s i o n , Magar suggested the use o f a c o n s t r a i n e d ( l e a s t squares method) o r i n f i n i t y norm to overcome the n e g a t i v e frequency problem. I f these methods f a i l e d , he recommended the use of ot h e r curve f i t t i n g procedures t h a t had p o s i t i v e c o n s t r a i n t s , o r the method o f Nelder and Mead (37). Use o f the m u l t i p l e r e g r e s s i o n approach as suggested by Magar c o u l d l e a d to a r a t h e r lengthy search u s i n g a l t e r n a t i v e methods i f f a i l u r e o c c u r r e d . T h e r e f o r e , the b a s i c c o n c l u s i o n t o be d e r i v e d from Magar's d i s c u s s i o n was t h a t m u l t i p l e r e g r e s s i o n c o u l d be u s e f u l i n o b t a i n i n g complete MWDs, but i n p r a c t i c e t h i s approach d i d not seem t o be f e a s i b l e . However, as Magar's concept o f u s i n g m u l t i p l e r e g r e s s i o n had m e r i t , an endeavor was made to overcome the l i m i t a t i o n s j u s t mentioned. S o l v i n g f o r the MWD u s i n g Magar 1s suggestion r e q u i r e d no changes i n the concept o r approach o r i g i n a l l y advocated by S c h o l t e . The o n l y d i f f e r e n c e between the two approaches was t h a t i n m u l t i p l e r e g r e s s i o n the squares o f the d e v i a t i o n s were minimized, r a t h e r than the a b s o l u t e d e v i a t i o n s . F o l l o w i n g the s uggestions made by Magar, the s o l u t i o n o f the MWD c a l c u l a t i o n u s i n g m u l t i p l e r e g r e s s i o n was t r i e d . The f i r s t attempts were focused on the data presented by S c h o l t e i n h i s o r i g i n a l a r t i c l e . A m u l t i p l e r e g r e s s i o n program s u p p l i e d by the Monroe c a l c u l a t o r company was used. However, i n i t i a l r e s u l t s d i d not a t a l l resemble those obtained by S c h o l t e . The problem v/as found t o be the presence o f an i n t e r c e p t term i n the m u l t i p l e r e g r e s s i o n s o l u t i o n . Since i t was not p o s s i b l e t o reprogram the Monroe c a l c u l a t o r to p r o v i d e a s o l u t i o n t h a t went through the o r i g i n , an a r t i f i c i a l method was d e v i s e d . Removal of the i n t e r c e p t was accomplished by e n t e r i n g an e x t r a dependent v a r i a b l e and a co r r e s p o n d i n g s e t of independent v a r i a b l e s , a l l of which were zeroes. T h e r e f o r e , the f i n a l m atrix to be so l v e d would appear as: c ( ? 1 ) / c 0 c ( ? 2 ) / c o C ^ n ) / c o 0 The addition of t h i s n u l l set at the end of the series of simultaneous equations to be solved e f f e c t i v e l y removed the intercept term i n the f i n a l s o l ution, by reducing i t s value very close to zero. Upon the i n c l u s i o n of the n u l l set, the analysis of Scholte's data allowed the re-production of the d i s t r i b u t i o n shown on page 116 of h i s paper (47). This p o s i t i v e r e s u l t was, however, short-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 resolved d i s t r i b u t i o n was investigated, using multiple regression, the negative frequencies noted by Magar (31) appeared. Since the a l t e r n a t i v e methods suggested by Magar were not r e a d i l y a v a i l a b l e , another approach was taken. The molecular weights that had produced negative frequencies were removed, on the premise that, i n r e a l i t y , they were not part of the so l u t i o n . The procedure can be i l l u s t r a t e d as follows: K l l K21 K31 K12 K22 K32 K ml K m2 Eq. 39 K l n K2n K 3n 0 K mn 0 Assume that f i v e molecular weights comprised a series used to f i t the MWD. If the t h i r d produced a negative frequency; f l M l f2 M2 " f3 M3 f4 M4 f 5 M 5 ' E q " 4 0 then the c a l c u l a t i o n would be repeated using the following: :1 M1 f2 M2 f4 M4 f5 M5 • E q ' 4 1 I f the removal of the molecular weight associated with the negative frequency produced a set of only p o s i t i v e frequencies, the c a l c u l a t i o n would then be moved onto the next s e r i e s . This series would be s h i f t e d by a factor 1/4 of 2 ' from the f i r s t , an i n t e r v a l proposed by Scholte. If t h i s was not the case, the negative molecular weight would again be removed, u n t i l only a p o s i t i v e solution was produced. The solutions obtained by using t h i s procedure were then compared to those obtained by Scholte"s l i n e a r programming method. I n i t i a l 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 solutions. However, i t was d i f f i c u l t to go through the complete c a l c u l a t i o n without errors. Therefore, i t was decided to program and automate a l l facets of t h i s c a l c u l a t i o n on the Monroe 1880 c a l c u l a t o r , to allow for easier assessment and comparison of t h i s approach. The r e s u l t i n g program required 4,000 program steps and almost a l l 512 data r e g i s t e r s available, taxing the c a p a b i l i t i e s of the ca l c u l a t o r to the l i m i t . The program allowed the negative molecular weights to be removed without requiring the r e c a l c u l a t i o n of the o r i g i n a l matrix and removed much of the tedium of the ca l c u l a t i o n . Using t h i s program, i t was found that the removal of negative molecular weights allowed reproduction of the results obtained by Scholte's program, i n the simpler cases. When two or three negative values appeared i n the solution rather than one, the choice of the order i n which the negative frequencies were removed led to d i f f e r e n t solutions. Through t r i a l and error, i t was found that the removal of the largest negative frequency produced s i m i l a r results when compared with Scholte's method. Although t h i s program was much easier to run than the o r i g i n a l , i t was s t i l l very time consuming, requiring approximately one hour or more to complete a ca l c u l a t i o n . I t became apparent that the Monroe version of the multiple regression program was able to produce comparable d i s t r i b u t i o n s to those obtained from Scholte's l i n e a r programming approach. Due to the time consuming nature of the ca l c u l a t i o n , a FORTRAN version was developed for use with the U.B.C. IBM 37 0/168 computer, i n order to reduce the c a l c u l a t i o n time and to allow more routine t e s t i n g . 68, During the investigations with the Monroe 1880 program, i t was observed that the s t a t i s t i c a l parameters were very-se n s i t i v e to the f i t of the chosen molecular weight s e r i e s , when the molecular weights were coincident with the maxima of the d i s t r i b u t i o n s . This led to an i t e r a t i v e concept that used the s t a t i s t i c a l parameters from the regression as a guide to f i n d i n g the correct molecular weights and concentrations of the components. A further expansion of the dummy variable concept, used i n removing the intercept from the o r i g i n a l m u l t i p l e regression program, was developed to smooth 'undefined' MWDs. Both of these concepts, o r i g i n a l l y noted i n the Monroe version, were included i n the FORTRAN program. Some of the r e s u l t s discussed i n the next sections were o r i g i n a l l y obtained: using the Monroe version, but could be more r e a d i l y i l l u s t r a t e d i n the context of the completed FORTRAN program. RESULTS AND DISCUSSION Model Systems In order to study and compare the l i n e a r programming and multiple regression approaches, and to 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 to the study of proteins, data were generated to represent error free models for study. One approach used to produce the model data was that of Scholte (48), assuming a log. normal d i s t r i b u t i o n , and the other was a simple expansion of the Rinde equation that did not assume a d i s t r i b u t i o n . A. Log normal'distribution method In his study, Scholte assumed that a log normal d i s t r i b u t i o n described the molecular weights of synthetic polymers. The choice of the d i s t r i b u t i o n function was a r b i t r a r y , and i n Scholte's work the log normal function was convenient, since the molecular weight series he used to describe the systems was l o g a r i t h m i c a l l y spaced. Using Scholte's approach, a program was written to generate log normal d i s t r i b u t i o n s . The basic log normal function can be represented as: where: S.D. - i s the standard deviation of the d i s t r i b u t i o n - i s the molecular weight associated with the maximum frequency of the d i s t r i b u t i o n f (M) exp Eq. 42 M i s the variable molecular weight f (M) the frequency obtained 70. Equation 4 2 r e p r e s e n t s a unimodal d i s t r i b u t i o n . In order to produce a multimodal d i s t r i b u t i o n , a s e r i e s of these f u n c t i o n s w i t h d i f f e r i n g maxima would 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 i s r e p r e s e n t e d by the f o l l o w i n g r e l a t i o n : f (M) = C 1 R-L exp -S.D.. I n M + Eq. 43 R 2 exp^- l n S.D, M M, + R 3 exp S.D.. l n M M'_, where: R^_3 are the p r o p o r t i o n s of each mode of the dis-*:;'. t r i b u t i o n S.D.^_3 are the standard d e v i a t i o n s o f each mode M l - 3 a r e t 3 i e m ° l e c u l a r weights a s s o c i a t e d w i t h the maximum frequency o f each mode C c o n s t a n t f o r v a r y i n g the sum o f the d i s t r i b u t i o n The above equation was programmed t o a l l o w the g e n e r a t i o n of f(M) a f t e r the p r o p o r t i o n s , means, and standard d e v i a t i o n s of the components had been d e f i n e d . T h i s program was made a subroutine o f a p l o t t i n g program so t h a t the d i s t r i b u t i o n c o u l d be p l o t t e d as i t was c a l c u l a t e d . The inp u t , output and p l o t of a thr e e 71. component system are presented i n Figures 13 and 14. In order to convert the d i s t r i b u t i o n data into the desired experimental values, a second program calculated values from the following expansion of the Rinde equation. c(5) A.M .exp(-X.M £) X M_ exp(-X .M0?) = - l - i ^ — — — f (M,) + --—£ ^-JL F ( M ) c 1 - exp(-A.M 1) 1 1 - exp(-X_.M„) o j 1 r j 2 X.M exp(-X.M O j n • j n 1 - exp(-X . M ) ^ j n f(M ) Eq. 44 The speed, p a r t i a l s p e c i f i c volume and r a d i a l distance i n t e r v a l were chosen, and the molecular weights and f(M) obtained from the above program entered to generate the 5 and c(5)/c data. The input and output of t h i s program i s i l l u s t r a t e d i n Figure 16. B. Simple expansion of the Rinde equation An a l t e r n a t i v e method to the log normal d i s t r i b u t i o n was the use of a simple expansion of the Rinde equation. Rather than using a d i s t r i b u t i o n to describe the system, dis c r e t e molecular weights were entered into the Rinde equation, along with the r e l a t i v e r a t i o s of the components, The Rinde equation was solved for c(£)/c according to the following function: Figure 13. Input and output of the log normal d i s t r i b u t i o n program. Input: A - p l o t t i n g instructions B - molecular weight of component 1 - standard deviation of component 1 - f r a c t i o n a l r a t i o of component 1 C - same as B for component 2 D - same as B for component 3 E - more p l o t t i n g instructions Output: F - l n molecular weight - f(M) 0 < • 0 0 0 • 0 0 1 0 i 0 0 2 • 0 0' 0 « • 0 0 0 • 1 7 0 • 0 0 4«- 0 0 2 « 0 0 0 • 0 0 C ) 0 5 0 • 0 0 0 • 5 0 A 4 • 1 7 0 • 0 0 0 • 0 1 o . 0 2 0 • c ) 0 5 4 « 3 4 1 0 ' 0 0 0 • 2 0 1 0 • 0 0 1 • 0 0 4 « 5 1 0 « - 1 0 0 • - 1 0 2 5,0 0 0 ' • 0 0 n 4 • • 6 9 0 « . 1 0 B 0 • » 0 2 0 < « 3 3 J 4 -. 8 6 8 0,0 0 0 < - 0 0 0 • 2 2 0 1 0 C 0 » 3 3 j 5 - 0 3 0 » 0 8 3 2 0,0 0 0 . 0 0 n 0 • 1 0 D 5. • 2 1 0 » 3 3 j 0 • ( 3 0 4 0 • 7 2 5 • 3 8 4' • 0 0 0 • 1 1 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 25,000-80,000 dalton mixture (1:1 ratio). Multispeed data; interval =2.0. Figure 16. Input and output of the K vs. c ( ? ) / c o program for the log normal d i s t r i b u t i o n . Input: 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 to the meniscus - number of entries B - l n molecular weight - frequency C - solution meniscus - c e l l bottom - s t a r t i n g p o s i t i o n - delta x - number of output values - rpm Output: D - chart p o s i t i o n - K ~ cU)/c_ E - X 76. 1 2 3 4 • 5 • 6 • 6 • 0 * 7 0 0 | .0 ^ J 2 0 • 0 A8 5 4 * 1 7 0 0 1 4,7 9 1 « 0 * 0 1 0 0 4 . 3 7 0 0 2 3,4 4 2 • 0 * 2 0 0 0 4 * 5 1 0 0 3 2,3 5 9 • 0• 10 0 0 4 * 6 9 0 0 4 6,9 7 7 . 0 • 0 2 0 0 1 B 1 0 1 1 1 2 6 2 * 3 3 3 3 8 5 * 0 0 0 0 6 2 * 3 3 3 3 5 * 6 6 6 6 4 * 0 0 0 0 1 2,0 0 0 • 0 0 6 2 * 3 0 0 0 0 14 5 9 6 7 * 9 7 5 5 3 2 0 6,8 7 3 • 6 5 0 7 1 2 9 6 9 7 9 * 3 2 5 5 3 4 3 2 1 1 D J 5 * 4 4 2 3 0 0 0 0 0 - 0 5 77. c U ) A .ML exp(-X.M £) X.M exp(-X.M O = _ J ± ! J i f + _ J _ J—± f c 1 - exp(-X.M ) 1 1 - exp(-X.M ) 2 O J x ] /• X.M exp(-X M J-^ f Eq. 45 1 - exp(-X .M.) 3 3 T h i s equation d e s c r i b e s a d i s c o n t i n u o u s d i s t r i b u t i o n r e p r e s e n t e d by three components i n some f r a c t i o n a l r a t i o t o each o t h e r ! T h i s approach pr o b a b l y r e p r e s e n t s the s t a t e o f p r o t e i n s i n s o l u t i o n , s i n c e p r o t e i n s are made up of d i s c r e t e molecular weights r a t h e r than having the wide range p r e s e n t i n s y n t h e t i c polymers. The i n p u t and output f o r t h i s program are presented i n F i g u r e 17. Both of these approaches were used t o generate model systems f o r the study of 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 the programs used i n t h i s work. The Rinde eq u a t i o n was favoured somewhat due to i t s s i m p l e r and f a s t e r o p e r a t i o n . 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 R e g r e s s i o n A t t h i s p o i n t , i t i s p e r t i n e n t to d e s c r i b e some of the b a s i c s of the FORTRAN v e r s i o n o f the MWD c a l c u l a t i o n , Figure 17. Input and output of the Rinde equation program. Input: A - c e l l bottom - meniscus - number of data points desired B - A - number of components Output: C - £ - c(K)/cQ 7 • 2 0 0 0 0 0 0 0 6 • 8 0 0 0 0 0 0 0 4 • 0 0 0 0 0 0 0 0 0 • 1 0 0 0 0 C 0 0 5 • 6 0 0 0 0 0 0 0 0 • 0 0 0 0 5 0 7 9 2 • 0 0 0 0 0 0 0 0 2 5,0 0 0 « 0 0 0 0 0 8 0,0 0 0 • 0 0 0 0 0 0 • 5 0 0 0 0 0 0 0 0 • 5 0 0 0 0 0 0 0 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 8 9 4 o • 2 5 5 3 5 7 1 4 1 . 3 7 0 8 0 8 5 3 0 • 5 0 7 1 4 2 S 5 0 • 7 2 7 0 0 2 8 7 o. 7 5 5 3 5 7 1 4 0 • 4 3 4 3 8 5 6 8 80. summarize some o f the main s u b r o u t i n e s and i l l u s t r a t e the output p r o v i d e d by t h i s program. The program was w r i t t e n w i t h the comparison of the r e s u l t s t o those of S c h o l t e 1 s l i n e a r programming program i n mind. T h e r e f o r e , the i n p u t formats were made i d e n t i c a l so t h a t any data c o u l d be r e a d i l y interchanged to a l l o w c a l c u l a t i o n by e i t h e r program. The program was composed o f a number o f major sub-r o u t i n e s t h a t w i l l be b r i e f l y d e s c r i b e d : M o l e c u l a r Weight ma t r i x A two d i m e n s i o n a l matrix m x n c o n s i s t i n g o f a l l the m o l e c u l a r weights r e q u i r e d f o r the s e r i e s t o cover the range of s o l u t e s was generated from a s i n g l e s t a r t i n g m o lecular weight. The i n t e r v a l , number of m o l e c u l a r weights (m) and the number of c y c l e s r e q u i r e d f o r the c a l c u l a t i o n (p) were i n p u t v a r i a b l e s . R e g r e s s i o n m a t r i x The m a t r i x of v a l u e s r e q u i r e d t o p r o v i d e the s e t of overdetermined simultaneous equations was c a l c u l a t e d a c c o r d i n g t o E q u a t i o n 34. The £ and c ( £ ) / c o v a l u e s were pr o v i d e d by the i n p u t data s e t . The m o l e c u l a r weights r e q u i r e d f o r the K^j term were drawn from the p r e v i o u s l y c a l c u l a t e d m o l e c u l a r weight m a t r i x . The m a t r i x generated i n t h i s manner was the complete m a t r i x r e q u i r e d 81. for a l l p cycles of the c a l c u l a t i o n . A vector, obtained from input data, was added to the k^j matrix to produce the actual simultaneous equations required to undergo multiple regression analysis. The t o t a l dimensions of the f i n a l regression matrix was p(m x n) and can be i l l u s t r a t e d as follows: c ( q ) / c o K n K 2 1 K 3 1 c(£ )/c K, K. K_ n o In 2n 3n cU1)/oQ K n K 2 1 K 3 1 c(K )/c K, v v n o l n K„ K_ 2n 3n K ml K mn K ml K mn Eq. 4 6 This large matrix was then converted into a single vector and put on f i l e to enable accessing by the multiple regression routine. ' Multiple regression The multiple regression program originated from the S c i e n t i f i c Subroutine Package developed by the IBM 82. C o r p o r a t i o n ( 3 ). I t was s u i t a b l e f o r the MWD c a l c u l a t i o n because i t was s p e c i f i c a l l y designed t o be used as a s u b r o u t i n e . The r e g r e s s i o n r e s u l t s i n c l u d e d an i n t e r c e p t , as d i d the Monroe program, and a n u l l s e t was generated to remove i t . 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 was i n c o r p o r a t e d i n t o the program to work on each o f the p p o r t i o n s of the t o t a l r e g r e s s i o n matrix (Equation 46), as d i c t a t e d by the appearance of negative f r e q u e n c i e s i n the s o l u t i o n . P o s i t i v e 'constraint A s m a l l s u b r o u t i n e was used to scan f o r the l a r g e s t n e g a t i v e frequency r e s u l t i n g from the m u l t i p l e r e g r e s s i o n c a l c u l a t i o n , and t o l o c a t e the K.. v e c t o r t h a t c o n t r i b u t e d -D to the n e g a t i v e v a l u e . T h i s s u b r o u t i n e c o n t r o l l e d the m u l t i p l e r e g r e s s i o n c a l c u l a t i o n i n i t s p r o g r e s s i o n through the p p a r t s of the r e g r e s s i o n m a t r i x . When a n e g a t i v e number was d i s c e r n e d , the address of the a s s o c i a t e d K.. v e c t o r was noted, and the v a l u e s i n the f i l e ID p e r t a i n i n g to a l l those were e f f e c t i v e l y removed by s h i f t i n g the n e i g h b o r i n g v e c t o r i n t o the l o c a t i o n o f the n e g a t i v e v e c t o r as f o l l o w s : 83. c ( ? 1 ) / c o K n K 2 1 K 3 1 K 4 1 " ' ' * Eq. 47 c(£ )/c K, K_ K_ K n o l n 2n 3n 4n If the K 3 1 vector produced a negative frequency, the new matrix becomes: c ( ^ ) / c o K n K2]_ K 4 1 Eq. 4 8 c(£ )/c K. K ' K. s n o In 2n 4n The dimensions of the section of the regression matrix to be analyzed by the multiple regression routine were then reduced appropriately and the regression repeated. This process was repeated u n t i l only p o s i t i v e frequencies were obtained. When t h i s was the case, the next region • of the regression matrix underwent sim i l a r treatment u n t i l a l l p cycles were completed. The r e s u l t i n g p o s i t i v e frequencies obtained from each cycle were used to produce the molecular weight d i s t r i b u t i o n . An example of the output of the FORTRAN program written to produce the complete MWD using multiple regression can be seen i n Figure 18. Some comments pertaining to various features of the output follow. X- i -v i x x i • r ( n i c » » m n v i . i :Lv . ; , : ^ 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 ^ ^ ^ > / > A \ f i t n': I i ,•„• . . " t f . . . 1 . . I M . F . . . 3 . . T I L ! : . . . J , . T l L f . I H . . . « . . T l L t . . , f r . . l t l . f . . . 7 . . T I L h . . . 0 . . l ! I f : ' . . . 1 . . l i Ll- 1 11.' . . . 1 . . ' J l s 1 i >.'.,. V . i-f.S V I ! . " ' OMVfcMSlTY UF H C CONFUTING CE^IKE T 3 ( A i< ] ? / ) 0/!cJj:u<> T'-.J » » * ? > / / ! F i g u r e 18. Example o f the output o f the m u l t i p l e regression -T4KD program. T T T T T T T T T T T T 1 ,"i n M M M M . _ . . . . . T T i r rT r r T T T T 011 i' f 1 f 1' nH i if c ; t ; | M M M f <M r- y, M nn ou M ?1 '-1M (,«!.# M K M M ^ T T ni'. Oil ' ! , • M I 'M M M Ms- S M \i>J ' T O f f no M M M * M M M M (.1 V! M M M M M T I nr> fi'i ?!M : i M M (; M '•• T I . (Hi no M *•. ?•! M « . - . -T T r)i« r "* (J S' M M M MM 7 T OP un M M W -1 M >1 T T n . » i n H >; IL. .1 M ' ! * M •: T r nnr<(VH >iH OC-lO M M M>' c r cere r.rxc-cc rcc cc cc •:c -ce-re cc CC cere :cr! cr cr. rr. rfrrrcrr crrcecc -Lt-1 T I M T-1-! I M T U I I I I 1 1 T ! ! ! I T U t I I I i t 1 1 1 T u 1 1 1 r i n i i nu ? w v v LI-LL LL / vv / V ' / V v v W . . . W W w I.LLLL'.U.l LLL I.LL'.t.LH LL1.L t . l A S T S T G , « ! l t < . H * S » 07t«9l.<U ,is£S> "Ie.<-<" StC.'ifP- CC »t 07] rmj A P 4 ?\ m Nil " ' 1 - 6 " IT .",:,T\ 5 >,.u-,i-c 3 r CtCl.f? ..r*i.cuL«rION M C B E « H F « E : •>:-.<C C A L O ' L A T J t ' N 't\t)f. 1 = T T T t W A T ftlN f-ODt. ? = SI<OOTHNI'IG . MCOF. THE P f " ? T tNA'.'T P . ' . ' J j*rTK'>S F0!» T H J S C A L C ' I L A T I ' I M A ' ) F ! CEI 1. ITT":' „T . an ii.n n- - l i F N T S F t ' S P A ' Si'ECt'-MC vol. 1 , ( J r t ( l ( 1 1 85. n 86. - <r o if c *•"» o * -O O IT -3 O 3 C f\J — U"> — a. o c =1 -c — c ~ ^  f Z UJ <> 7) 3 O C I 1 * O A) iirtj : £ j ^ O O A. a u. rv o ru o rr — o o c c: s c c = e c LOI 3 ^ O -t «* U- c »- -» OUJ < > •AJ M ia tt a > c Ou. MJ <4 * a ^ ; u o u. C 13 •* 4 »- UJ lfl UJ n o cn i -1 ul -A O Z O- ftj 4> C C-WOK C "O o — C — O C r*> ff C IT S Ui U. bit ^ Q tr a. a u. c 3 u; W o « u. 3 •Ji — »- z < _• r < '.2 > ; r- ui v j • • J : i o u. — -* = = c C C 3 o o I c tt « . u. z »- ' o « * Z UJ 5' O *-a »-» •» UL O »- — c w •> > UJ UJ a a > c a o u. ui UJ a c c a i u o i i u; o * • - c o -* 7, r<- ut tr. U J « * * * THE SFRIES IS- H a t ' . 33000. 56000. " S i s ' * . TMF FREQUENCIES -S-.P. OF FBCnilEnC TFS I S « 0.0J8<>5 0.U2236 0.S2U88 0,27660 0,63006 0.Ont, 8.7 0.01 U13 B . l l t M n^m^u » , » » » » Slix O F FREQUENCIES f S - 1.89285 TA«LF f)F STATISTICAL PARAMETERS » = a = - T g r - " ^ i r : - - - T I M T E R C F P T -ST(1 EHROI) OF E S T DEC. OF FI)E t *0M ss OF oEviATiriM rpo« B F C M Mg»N SM PE-V1AT lUtlS . 0.00102. 5'. 0 0 0 0 ll 0 . 00002 -()'. onaau-HLILT CORK COEF S U M S U OUE TO REGN_.-M F A N S O DF S S . DEG UK FREEDOM FwVAl tIK 1.00000 11.0 1362 . 2.8027? 1 o. no (mo ?To'i u i . o n END OF.... CYCLE NUMBER 2 . THE M.ri S F K I E S I S - .. . THE FREQUENCIES ARE* . 13038.. 0,0761 I _ 22165,. 0,•5(1616 .. 37681. 0 . 3 2 3 0 5 . 6<I058. o ,<ioi sa 108808. o.a ? 3 ? o S . n , O F F R F M I E N C I E S i s - o.oo?06 . SUM. OF F R F . Q U E N . C I E S I S - . . . " . 1.900711 . O.0O6S0 0.00718 0.00587 0,00271 • TABLE Of STATISTIC*! PAUAMFTFRS . INTER.CE.PT . : S T D F R P 0 8 nF E S T D E P OF F R F F O O M . n nr nFvjfATTOM panu prr.. H E AN S O OE V I A T I O N S 3.00.016. O.oon/iO 5'. 0 0 o f, o ll'. il 0 0 (10 .. .H'JLT C O R K . C O E F SU* Su. DUE TO REGN MEAN SU OF SS 0 F « OF FHFFOOM o.ooooo F - V A L U E ... l.O"o?'j l i . o t 2.fi027S 19.PP nop .1 1 9l)5 J3<l .54 ENfi OF CYCLE NUrtBEH 3 l l l l l l l l l l l . l l . l . . . . . . . . . » . > . » l . > l . l l < . | . 1 | | t | ) l | < ) . , | H l l l ( ' l < l t l ) l t > > U » t > l > » l | » ( < l | l » » l l « l < l l > » I X l | H I I H I H I I < l T H E Hn SFPIF3 IS-TrtF F9"f.flnFpiiC IFS A HE. I AflBB. 25310. 0 . 1 7533 n.hPStlT . (13026. 0.2161 I 731 (15 . n . t i o i 31 I21306, P.2567lV. S . n , O F F R E O H E N C I F S I S . S U M O F F R F O M E M C I E S I S -0 , 0 0 3 0 5 o,oo6sn n . o o n i i 0,00679 0 , 0 0 3 1 0 I.90/108 TAPLF |iF STATISTICAL PARAMETERS 88.. i c. -c r- c -C K l r t C 3 ' c *- c e x c e a c- c I • • i j i < 2 ! ! U 2 U. £} »- -e u£ •« > : »- a a :> c • a C' u u_ i w a | c a j cr u: O u. > c d < l 7 f- U,' « w i «n c x 89. 1. A table of pertinent parameters was printed to define the variables used i n the c a l c u l a t i o n . 2. The raw data used for the c a l c u l a t i o n , X ,£ and c ( ^ ) / c o were printed for reference and checking purposes. A fourth constant was also printed to i d e n t i f y whether the data were from schlieren or UV optics, so as to process the data accordingly. The data as presented were i n the same format required for Scholte's l i n e a r programming program. A zero set was always put i n at the end of the data set to allow the generation of the dummy variables which e f f e c t i v e l y removed the intercept. 3. For convenience, the i n d i v i d u a l molecular weight values pertinent to each regression were printed. 4. The frequencies, or the regression c o e f f i c i e n t s , pertaining to the corresponding molecular weights i n the series were printed. 5. The sum of the regression c o e f f i c i e n t s was printed, because i t was a valuable indicator of whether the chosen series was correct. 6. A table of s t a t i s t i c a l parameters allowed a judgment of the f i t attributed to the series used i n the regression. 7. If a negative frequency appeared, the location of the largest one was printed. This location was always i n r e l a t i o n to the number of molecular weights used i n the 90. ^egression. Therefore, the message "Molecular weight number 4 i s negative and w i l l be removed" referred to the fourth molecular weight i n the o r i g i n a l s e r i e s . If the next regression s t i l l produced a negative frequency, i t s loca t i o n would then be pertinent to the reduced series used i n the l a s t regression. This can be best i l l u s t r a t e d as follows: Assuming 7 molecular weights were used for the regression: f l M l f2 M2 f 3 M 3 f4 M4 ~ f5 M5 f 6 M 6 f7 M7 E q ' 4 9 f^Mj. was negative for the f i r s t regression and therefore removed. The next series to be regressed would be: f l M l f2 M2 f3 M3 f4 M4 f6 M6 f7 M7 E q ' 5 0 Therefore, i f a negative frequency occurred i n t h i s regression for f^-M-, i t would be l a b e l l e d as the removal D D of molecular weight 5. This f a c t should be kept i n mind when associating the po s i t i v e frequencies with the o r i g i n a l molecular weight seri e s . 8. Whenever the c a l c u l a t i o n had reached a p o s i t i v e solution, the cycle number p was printed, along with two l i n e s of stars to separate the conclusions of the i n d i v i d u a l cycles. 91. Comparison of Linear Programming and Multiple Regression Once the FORTRAN version of the multiple regression program was completed, a d i r e c t comparison of the two approaches was rea d i l y f e a s i b l e . From the analysis of a wide var i e t y of models (2 and 3 component) i t could be concluded that the two calculations produced almost i d e n t i c a l r e s u l t s . To make these d i r e c t comparisons, the molecular weight series, the i n t e r v a l and the range of each c a l c u l a t i o n had to be i d e n t i c a l . In order to i l l u s t r a t e the e f f e c t of the negative molecular weight removal and how i t led to almost the same soluti o n as that produced 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 Table III refe r s to the second regression of the K.. matrix that 1 3 was required any time a negative frequency appeared. Since any molecular weight associated with a negative frequency was removed from the regression, i t could be assigned a zero frequency value. In comparing the frequency values obtained by l i n e a r programming and by multiple regression, a very close agreement between the two solutions was observed. The analysis of more complex systems showed similar r e s u l t s . 92. T a b l e I I I . C o m p a r i s o n o f F r e q u e n c i e s R e s u l t i n g f r o m t h e 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 M o l e c u l a r W e i g h t ( d a l t o n s ) M u l t i p l e R e g r e s s i o n L i n e a r f (M). 1 f ( M ) 2 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 - 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 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 s p e e d d a t a Factors Af f e c t i n g the Molecular Weight D i s t r i b u t i o n Once i t became apparent that the multiple regression program could produce si m i l a r r e s u l t s to l i n e a r programming, a study was undertaken to investigate the e f f e c t s of speed, molecular weight i n t e r v a l and molecular weight range on the d i s t r i b u t i o n obtained. In his work, Scholte concluded that multiple speeds and a molecular weight i n t e r v a l of 2.0 were the most suitable c r i t e r i a for the analysis of polymeric systems. However, protein systems are comprised of discrete molecular weight components rather than continuous d i s t r i b u t i o n s . The simple Rinde expansion model i s rep-resentative of a mixture of discrete components, but analysis of data generated from t h i s model produced a continuous d i s t r i b u t i o n . Therefore, even an i d e a l mixture of homogeneous proteins, not subject to experimental error 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 could not resolve an i n f i n i t e l y narrow d i s t r i b u t i o n . Upon analysis of log normal model data, with various standard deviations, i t was found that the d i s t r i b u t i o n obtained from a standard deviation of 0.05 was e s s e n t i a l l y i n -discernable from that of the Rinde model. In general, broader d i s t r i b u t i o n s were easier to reproduce than narrower ones which were more demanding i n the correct choice of speed, i n t e r v a l and range. Since 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 of the variables on the d i s t r i b u t i o n . In the ensuing study i t became apparent that the e f f e c t s of the variables were not independent of each other, and interacted i n a complex fashion. A. ' Speed Speed was a c r i t i c a l factor i n the production of a well defined molecular weight d i s t r i b u t i o n . By applying increasing speeds, fractionation of the sample occurred, allowing the various components to contribute to the d i s t r i b u t i o n . At low speeds the higher molecular weights were the major contributors to the concentration 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 relegated to the bottom of the c e l l allowing the low molecular weight components to contribute. No problems-were encountered i n reproducing the d i s t r i b u t i o n of two and three component systems when a wide range of speeds (six speeds ranging from 10,000 to 30,000 rpm) were used to evaluate model data, using four data points per speed as advocated by Scholte. However, since more data points were r e a d i l y available from the UV scanner, a reduction in the number of speeds, with a concurrent increase i n data points was attempted. In general, i t was found that the number of speeds could be reduced to one speed per component, i f 20 data points were used and the speeds were chosen ju d i c i o u s l y . If the number of speeds chosen was i n s u f f i c i e n t or did not allow a s u f f i c i e n t 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 of a defined and undefined d i s t r i b u t i o n for a two component system can be seen i n Figures 15 and 19 respectively. The undefined d i s t r i b u t i o n had several features which made i t r e a d i l y recognizable. These were a f l a t topped and non-bell shaped peak for the lower molecular weight component and a s h i f t of the high molecular weight peak to a lower molecular weight than o r i g i n a l l y defined by the model d i s t r i b u t i o n . The term 'undefined' also carried additional s i g n i f i c a n c e i n that the true r a t i o s of the components were l o s t , and the low molecular weight component varied i n i t s l o c a t i o n on the x-axis, i f the range was changed. A s i m i l a r s i t u a t i o n existed i n the case of three component systems. In a l l cases, 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 present in the system, i f they were s u f f i c i e n t l y separated i n molecular weight (Figure 20). This property of single speed d i s t r i b u t i o n s was l a t e r used advantageously. 96. 0.8 0.6 S 0.4 0.2 r 0.0 © © © 0 t ,.Q. © 14 28 M x 10" 56 112 224 Figure 19. Senrilogarithmic p l o t o f f (M) vs. M f o r a 25,000-80,000 dalton mixture (1:1 r a t i o ) . Single 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 Figure 20. Sendlc>garit±imic p l o t o f f (M) vs. M f o r a 25,000-80,000-320,000 dalton mixture (1:1:1 r a t i o ) . Single speed data; i n t e r v a l =2.0. B. I n t e r v a l and range The e f f e c t of choosing an a p p r o p r i a t e m o l e c u l a r weight i n t e r v a l and range c o u l d be very b e n e f i c i a l i n o b t a i n i n g the b e s t p o s s i b l e MWD. S c h o l t e concluded from h i s work t h a t an i n t e r v a l o f 2.0 between the m o l e c u l a r weights was optimum. H i s assumption was found to be t r u e i n t h i s study, s i n c e the i n t e r v a l to be used was dependent upon the range of the m o l e c u l a r weight s e r i e s . That i s , i f the system under study covered o n l y a r e l a t i v e l y s h o r t range, the d i s t r i b u t i o n was 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 a s m a l l e r i n t e r v a l . The i n t e r v a l c o u l d be changed by a l t e r i n g the c o n s t a n t or the power v a l u e or b o t h t o s u i t the system. An example of the improvement which was p o s s i b l e by choosing the a p p r o p r i a t e i n t e r v a l can be seen by comparing F i g u r e s 15 and 21. F i g u r e 21 used the same dat a as F i g u r e 15 and an obvious improvement i n r e s o l u t i o n and shape c o u l d be observed when the i n t e r v a l was reduced from 2.0 t o 1.5. F i g u r e s 22 to 24 r e p r e s e n t a model o f the s e l f - a s s o c i a t i o n of bovine serum albumin (BSA) a t e q u i l i b r i u m i n a r a t i o o f 2:1. The f i r s t d i s t r i b u t i o n i l l u s t r a t e d how an i n t e r v a l of 2.0 allowed no r e s o l u t i o n of the two components * The next one, u s i n g an i n t e r v a l of 1.4 r e s o l v e d two peaks, but they were d i s t o r t e d . Adjustment of the range to s u i t the system more c l o s e l y 98. 1.0 0.6 V 0.2 M x 10 Figure 21. Semilogarithmic plot o f f CM) vs. M for a 25,000-80,000 dalton mixture (1:1 ratio). Multispeed data; interval = 1.5. 0.8 0.6 0.4 0.2 0.0 ' ~~ G 0 30 Figure 22. 60 © \ © 0 120 240 M x 10 -3 Semilogarithmic plot of f CM) v s . M for a 67,000-134,000 dalton mixture (2:1 ratio). Multispeed data; interval = 2.0. 30 42 . 58 82 115 161 225 M x 10~ 3 Figure 23. Semlcgarithrtdc p l o t o f f (M) v s. M f o r a 67,000-134,000 dalton mixture (2:1 r a t i o ) . Multispeed data; i n t e r v a l - 1.4. 0.8 ©• 0 \ © > / / / © \ \ © / N * • / . \ © \ 0 © • * / • • Q Q I I 2 . 30 42 58 82 115 161 225 -3 M x 10 Figure 24. Serrdlogarittaic p l o t of f (M) vs. M f o r a 67,000-134,000 dalton mixture (2:1 r a t i o ) . Multispeed data; i n t e r v a l = 1.4, with the range having 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 desired 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 correct proportions of the components along with the correct weight average molecular weight. C. Loss of data One other factor investigated was the e f f e c t of removing high c(£)/c values from the synthetic data. This had to be considered because at higher speeds the model C ( C ) / C q values rea d i l y attained much larger values than could be r e a l i s t i c a l l y evaluated by any o p t i c a l system. Removal of these values usually did not a f f e c t the d i s t r i b u t i o n , as long as there were enough other data points available. The only d i s t r i b u t i o n s affected by removal of c ( / c values were those involving single speed data derived from too high a speed. D. Summary• The d i s t r i b u t i o n s discussed were a minor part of the t o t a l number of variations and combinations studied i n order to understand how the variables affected the d i s t r i b u t i o n . The major conclusion to be derived from the r e s u l t s was that some pri o r knowledge of the molecular 101. weights of the components would be h e l p f u l i n d e t e r m i n i n g the c o r r e c t combination of speed, i n t e r v a l and range. Thus, i n a s s e s s i n g experimental data, a number of MWDs had to be c a l c u l a t e d to g a i n i n s i g h t i n t o the most a p p r o p r i a t e range and i n t e r v a l . Expanding the M u l t i p l e Regression Approach A. ' A s s e s s i n g the s t a t i s t i c a l ' parameters Up to t h i s p o i n t , no mention has been made o f the s t a t i s t i c a l parameters a v a i l a b l e through the use o f m u l t i p l e r e g r e s s i o n i n the MWD c a l c u l a t i o n . The a v a i l a b i l i t y o f these parameters was the major r a t i o n a l e f o r p u r s u i n g the concept put forward by Magar (30). He had suggested t h a t the s t a t i s t i c a l parameters would allow the e v a l u a t i o n o f the f i t o f any m o l e c u l a r weight s e r i e s t o a d i s t r i b u t i o n . B a s i c a l l y , t h i s turned out to be t r u e . I f the m o l e c u l a r weight s e r i e s chosen f o r a p a r t i c u l a r system d i d not cover the complete range, or was completely out o f l i n e w i t h the weights making up the system, a much poorer and v e r y n o t i c e a b l e r e d u c t i o n i n f i t was noted. However, when the m o l e c u l a r weights covered the complete range, f o r a l l p r a c t i c a l purposes l i t t l e d i f f e r e n c e c o u l d be d i s c e r n e d between a s e r i e s which b e t t e r s u i t e d the system and a 102. s e r i e s which was poorer. The sum of the f r e q u e n c i e s was a l s o found to be a good i n d i c a t o r of whether the range was wide 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 the range was i n s u f f i c i e n t and a c o n s t a n t being o b t a i n e d when the range was good. The major c r i t e r i o n of a good d i s t r i b u t i o n , t h e r e f o r e , became the MWD p l o t , w i t h the s t a t i s t i c a l parameters being used mainly to assess the c o r r e c t c h o i c e of range. B. The concept of an ' i t e r a t i v e s o l u t i o n Although the s t a t i s t i c a l parameters were h e l p f u l i n the c h o i c e of the c o r r e c t range f o r the c a l c u l a t i o n , t h e i r a c t u a l u s e f u l n e s s was somewhat d i s a p p o i n t i n g . However, d u r i n g the a n a l y s i s of the many model d i s t r i b u t i o n s c a l c u l a t e d , i t was noted t h a t i n some cases the frequency v a l u e s a s s o c i a t e d w i t h the m o l e c u l a r weights were d i r e c t l y p r o p o r t i o n a l to the r a t i o s of the i n d i v i d u a l components p r e s e n t . T h i s s i t u a t i o n appeared o n l y when the m o l e c u l a r weight s e r i e s was c o i n c i d e n t a l w i t h the maxima of the 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 . T h i s c o i n c i d e n t a l a s s o c i a t i o n w i l l be i l l u s t r a t e d i n the f o l l o w i n g example. Assuming t h a t model data had been generated f o r a mixture of two molecular weight components, one o f 20,000 and the o t h e r 80,000 d a l t o n s , i n a r a t i o o f 1:1, then a c o i n c i d e n t a l m o l e c u l a r weight s e r i e s would be: 103. 104. 10,000 20,000 40,000 80,000 and the coincident molecular weights would be 20,000 and 80,000 daltons. The frequency values obtained from the MWD c a l c u l a t i o n for these molecular weights would be 0.50 for both components. This was the case whether multispeed or single speed data were used. This observation implied that unispeed data, i f solved only for the two maxima rather than a series, would allow the quantitation of the r a t i o of the two components. In the case of multispeed data, a reasonable facsimile of the true d i s t r i b u t i o n could be obtained, that related to the r a t i o of the components d i r e c t l y by the r e l a t i v e area under the peaks. This was not possible with unispeed since an undefined d i s t r i b u t i o n was generally produced. By using the MWD program to solve for 20,000 and 80,000 daltons only, and terminating the c a l c u l a t i o n a f t e r one cycle, the two frequency values obtained were s t i l l both 0.50. The s t a t i s t i c a l parameters indicated an excellent f i t . Any other combination of molecular weights produced a s i g n i f i c a n t l y poorer f i t , as judged by the s t a t i s t i c a l parameters (F - r a t i o ) . Some res u l t s of t h i s c a l c u l a t i o n can be seen i n Table IV. Investigation of three component systems indicated that s i m i l a r quantitation from single speed data was possible. c 105. T a b l e IV. 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 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 M i x t u r e (1:1 R a t i o ) U s i n g S e l e c t e d M o l e c u l a r W e i g h t P a i r s , M o l e c u l a r W e i g h t F r e q u e n c i e s F - R a t i o P a i r 82,000 0.46 22,000 0.53 20,000 0.33 2.0 x 1 0 6 80,000* 0.50 7 8.6 x 10 20,000 0.50 85,000 0.42 5 2.0 x 10 25,000 0.56 85,000 0.48 3 7.7 x 10 16,000 0.53 71,000 0.63 4 1.7 x 10 16,000 0.37 65,000 0.67 3 2.8 x 10 * B e s t f i t s o l u t i o n 106. Two and t h r e e component model systems could, be q u a n t i f i e d by e n t e r i n g the c o r r e c t m o l e c u l a r weights i n t o the r e g r e s s i o n equation. T h i s f a c t , and the o b s e r v a t i o n t h a t the r e g r e s s i o n s t a t i s t i c a l parameters were good i n d i c a t o r s of the c o r r e c t m o l e c u l a r weight v a l u e s , l e d to the concept of u s i n g these parameters as a guide to 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 to d e s c r i b e a system. I n i t i a l l y , i t was d e c i d e d t h a t an i t e r a t i v e a l g o r i t h m based on the c h a r a c t e r i s t i c s of a . s i n g l e speed unde f i n e d d i s t r i b u t i o n c o u l d be used t o f i n d the b e s t f i t m o l e c u l a r weights, and hence t h e i r c o r r e s p o n d i n g r a t i o s . C. The i n i t i a l i t e r a t i v e a l g o r i t h m Undefined d i s t r i b u t i o n s had c h a r a c t e r i s t i c f e a t u r e s , one of which was the s h i f t of the weight average m o l e c u l a r weight to lower v a l u e s on the m o l e c u l a r weight a x i s . The o t h e r f e a t u r e was t h a t a peak was o b t a i n e d f o r each component p r e s e n t . These f a c t s were u t i l i z e d as a b a s i s f o r a simple i n c r e m e n t a l a l g o r i t h m t h a t allowed the c o r r e c t m o l e c u l a r weights to be searched f o r , u s i n g the s t a t i s t i c a l F - r a t i o as a guide. The b a s i c a l g o r i t h m w i l l be d e s c r i b e d i n p o i n t form, f o r a t h r e e component system. 107. 1. The weight average molecular weight was c a l c u l a t e d from the o r i g i n a l u ndefined m o l e c u l a r weight d i s t r i b u t i o n o b tained from a complete MWD c a l c u l a t i o n , u s i n g the f o l l o w i n g r e l a t i o n : y M . f. M = — Eq. 51 E f ± 2. The weight average molecular weight v a l u e s were run through one c y c l e of the r e g r e s s i o n c a l c u l a t i o n and the r e s u l t i n g F - r a t i o s t o r e d . The h i g h m o l e c u l a r weight (M^) component was then incremented by some v a l u e dM^ and r e g r e s s e d again. The F - r a t i o o b t a i n e d from t h i s r e g r e s s i o n was compared to t h a t of the p r e v i o u s v a l u e , and, i f g r e a t e r , the M^ v a l u e was incremented a g a i n . T h i s procedure continued u n t i l the F - r a t i o maximized. A f t e r maximization, the M2 was incremented by CLM2 u n t i l the F - r a t i o maximized again, then c o n t r o l was passed t o M^. A f t e r going through a l l three m o l e c u l a r weights, c o n t r o l passed back to the h i g h molecular weight component, the increment dM^ was h a l v e d and the a l g o r i t h m was repeated. As :;. expected, the molecular weight changes became v e r y s m a l l a f t e r a number of c y c l e s . In g e n e r a l , approximately twenty i t e r a t i o n s were r e q u i r e d t o allow convergence t o the b e s t f i t molecular weights. T h i s a l g o r i t h m was o r i g i n a l l y t e s t e d on the Monroe 1880 and proved t o be s u f f i c i e n t l y promising t o be programmed i n the FORTRAN 108. v e r s i o n f o r f u r t h e r t e s t i n g on the IBM computer. A v a r i e d assortment of models was t e s t e d and the a l g o r i t h m was found to work reasonably w e l l i n most cases. T h i s i n i t i a l a l g o r i t h m served to e s t a b l i s h the f a c t t h a t a search method was a f e a s i b l e approach to s o l v i n g f o r the maxima o f unde-f i n e d d i s t r i b u t i o n s , from which s e m i - q u a n t i t a t i v e frequency v a l u e s c o u l d be o b t a i n e d . D. The simplex i t e r a t i v e a l g o r i t h m Although the u n i v a r i a t e i t e r a t i o n procedure d e s c r i b e d i n the p r e v i o u s s e c t i o n allowed convergence to good e s t i m a t e s of the t r u e frequency v a l u e s , i t had two major l i m i t a t i o n s . . The f i r s t was t h a t the search was u n i d i r e c t i o n a l , and the second t h a t the v a r i a b l e s were not independent. Furthermore, the m o l e c u l a r weight increment had t o be chosen w i t h care s i n c e too l a r g e an increment c o u l d overshoot the optimum and too s m a l l might never reach i t . I t was d e c i d e d t h a t a m u l t i v a r i a t e approach would be a b e t t e r method o f s o l v i n g f o r the f r e q u e n c i e s o f undefined MWDs. In r e c e n t y e a r s , m u l t i f a c t o r i t e r a t i v e o p t i m i z a t i o n procedures have been developed, b u t have n o t been used f r e q u e n t l y . M u l t i f a c t o r a n a l y s i s was i n i t i a l l y proposed by Box and Draper (5), who developed the e v o l u t i o n a r y o p e r a t i o n s procedure (EVOP) of s e a r c h i n g f o r the optimum by u s i n g a combination of f a c t o r i a l d e s i g n and r e g r e s s i o n t e c h n i q u e s . 109. A major advance over the EVOP was the simplex optimization method developed by Spendlay et a l . (51). This was an empirical i t e r a t i v e feedback c a l c u l a t i o n that was i n t u i t i v e l y simple i n concept and operation. This o r i g i n a l method was improved by Nelder and Mead (37) and further modified by Morgan and Deming (33). The simplex optimization had a unique advantage over the single factor approaches because of i t s a b i l i t y to overcome the ridge phenomenon that could prevent optimization due to variable i n t e r a c t i o n . 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 for the optimum could be instigated, with a l l the variables changing simultaneously. 1. The simplex method The simplex procedure has been described i n several publications, with the most e a s i l y grasped version presented by Morgan and Deming (33). However, even t h e i r description lacked c l a r i t y as i t did not show a simple example c a l c u l a t i o n . In the following pages, the concept, operation and c a l c u l a t i o n of the simplex w i l l be presented i n r e l a t i o n to i t s use i n searching for the optimum of an undefined MWD. Once the procedure has been firml y demonstrated, a flowchart w i l l best i l l u s t r a t e the r e p e t i t i v e nature of the r e s t of the algorithm. 110. The simplex concept can be represented by a geometric figure composed of one more dimension than the o r i g i n a l number of factors under inves t i g a t i o n . Thus, for a two variable system the simplex i s represented by a t r i a n g l e , for a three variable system, a tetrahedron, and so for t h . The vert i c e s of these geometric figures indicate the response (dependent variable) to the contributing factors. For i l l u s t r a t i v e purposes a two factor system w i l l . . be discussed. Referring to Figure 25, the t r i a n g l e F^ represent the F-ratios (response) obtained by regressing the three sets of two molecular weights (f a c t o r s ) . These F-ratios are the basis for determining successive simplexes, thereby propagating a search for the best f i t combination of molecular weights. Ignoring at t h i s time how the o r i g i n a l factors were chosen, the molecular weights were put into matrix form as follows: F b F n Fw r e P r e s e n t s t n e s t a r t i n g simplex for an optimization search of a two component system. The v e r t i c e s F. F , and 10,000 60,000 regression :. F w 29,319 67,765 Eq. 52 15,17 6 88,97 8 F n 0 L 50 60 70 80 90 M x 10 -3 100 110 120 Figure 25. I l l u s t r a t i o n o f the f i r s t basic operations of the simplex optimization routine. 112 . The simplex a l g o r i t h m r e q u i r e d t h a t t h i s s t a r t i n g m a t r i x , r e p r e s e n t i n g the v e r t i c e s o f the s t a r t i n g simplex, would undergo s p e c i f i c geometric changes of r e f l e c t i o n , expansion and v a r i o u s forms of c o n t r a c t i o n based on the e v a l u a t i o n of the response ( F - r a t i o ) . The F - r a t i o v a l u e s F , F, and c ' w b F r e f e r r e d to the worst, b e s t and next t o b e s t v a l u e s n r e s p e c t i v e l y . The i n i t i a l s tep i n the simplex a l g o r i t h m was the c a l c u l a t i o n o f the c e n t r o i d , the mean o f the more r e s p o n s i v e f a c t o r s : C = (B + "N)/2 E q . 53 where C - c e n t r o i d B - b e s t m o l e c u l a r weight N - mean of a l l m o l e c u l a r weights i n the v e c t o r , other than the be s t and worst. t h e r e f o r e : C± = (67 ,765 + 8 8 , 9 7 8 ) / 2 = 7 8 , 3 7 1 C 2 = (29 ,319 + 1 5 , 1 7 6 ) / 2 = 22 ,247 The c e n t r o i d would then be l o c a t e d a t C^, C^. These c e n t r o i d v a l u e s were used t o generate a r e f l e c t i o n o r m i r r o r image of the worst l o c a t i o n , a c c o r d i n g t o the f o l l o w i n g r e l a t i o n : R = C + (C - W) Eq.54 113. where W r e p r e s e n t s the worst mo l e c u l a r weight o f each s e t . Th e r e f o r e : The r e f l e c t e d v a l u e s occupied the p o s i t i o n R^, R 2 and re p r e s e n t e d a new molecular weight s e t t h a t c o u l d be re g r e s s e d t o o b t a i n a new response F . I f the F v a l u e was g r e a t e r than F^, an expansion was performed a c c o r d i n g to the f o l l o w i n g r e l a t i o n : The expanded v a l u e s became E-^, E 2 and the r e g r e s s i o n o f these m o l e c u l a r weights produced F . I f F g was g r e a t e r than F r then m o l e c u l a r weights F^, r e p l a c e d the worst response molecular weights i n the matrix as f o l l o w s : R± = 78,371 + (78,371 - 60,000) = 96,742 R 2 = 22,247 + (.22,247 - 10,000) = 34,494 E = C + 2 (C - W) Eq. 55 Th e r e f o r e , E 1 = 78,371 + 2(78,371 - 60,000) = 115,113 E 2 = 22,247 + 2(22,247 - 10,000) = 46,741 46,741 115,113 29,319 67,765 Eq. 56 15,17 6 88,978 114. This matrix was used to calculate a new centroid and the entire process repeated. The r e f l e c t i o n and expansion values of the o r i g i n a l simplex did not always exceed the best response, and for these si t u a t i o n s a set of contraction operations existed. These four contraction operations are presented, along with conditions of usage i n Table V. At t h i s point, i t becomes more fe a s i b l e to look at the algorithm i n flowchart form as presented i n Figure 26. The flowchart w i l l i l l u s t r a t e the re p e t i t i o u s nature of the r e f l e c t i o n , expansion and contraction operations, each operation providing an opportunity to produce a better set of factors than the previous worst set. These operations o s c i l l a t e back and fo r t h u n t i l the solution comes close to the optimum where no s i g n i f i c a n t l y better response can be obtained. 2. The s t a r t i n g matrix At the beginning of t h i s discussion, the formation of the o r i g i n a l simplex was ignored. There was a s p e c i f i c procedure advocated by Spendlay et a l . (51) for the production of the s t a r t i n g matrix, based on the number of variables to be optimized. Their method required that a range be chosen f o r each of the variables and that these variables 115. T a b l e V. 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 O p e r a t i o n / F o r m u l a U sage C o n d i t i o n R e s u l t i n g F - R a t i o (1) R e f l e c t i o n c o n t r a c t i o n F n > F r < F b F r c R = C + 0.50 (C - W) c (2) M a s s i v e r e f l e c t i o n -c o n t r a c t i o n F r c < F r F m r c R = C + 0.25 (C -W) mc (3) W o r s t c o n t r a c t i o n F r < F w F c w C = C - 0.50 (C - W) w (4) M a s s i v e w o r s t c o n t r a c t i o n . F <F F ' cw w mew C = C - 0.25 (C - W) mw 116. be : regularized by a l i n e a r transform to range from zero to one. The transform i s as follows: y = x - R m i n dR Eq. 57 where x mm dR - any measured value within the range - regularized value - lowest value of the range covered - the absolute difference 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 differences i n units and orders of magnitude of the variables. Spendlay's regularized matrix from which the response simplex was formed would be as follows for k variables: 0 0 P q q p o q q Eq. 58 117. where: — - { (k - 1) W k + 1 } kV2 1 ' Eq. 59 q = k + 1 Eq. 60 The s t a r t i n g matrix presented at the beginning of the simplex algorithm discussion was derived for a 20,000 and 80,000 dalton mixture, with a defined range of 10,000 to 30,000 for the f i r s t component, and a range of 60,000 to 90,00 0 daltons for the second component. Spendlay's regularized matrix i n that case would be: 0.00 0.96 0. 28 0.00 0.26 0.96 Eq. 61 The values i n the matrix were the s t a r t i n g regularized values that, i f converted back to molecular weights by reversing the re g u l a r i z a t i o n transform (Equation 57), produced the s t a r t i n g matrix shown at the beginning of the discussion. 118. 3. The simplex output The simplex a l g o r i t h m presented i n F i g u r e 26 was programmed i n FORTRAN ( 3 2 ) , i n i t i a l l y as a separate program t o al l o w the t e s t i n g and e v a l u a t i o n o f the a l g o r i t h m . Through t e s t i n g a g a i n s t manual c a l c u l a t i o n s , the r o u t i n e s were shown t o be working, and the simplex r o u t i n e was i n c o r p o r a t e d i n t o the complete MWD c a l c u l a t i o n and t e s t e d "with model systems. An example of the begin n i n g o f the output of the simplex i t e r a t i o n can be seen i n F i g u r e 27. I n i t i a l l y the output was s i m i l a r to the normal MWD c a l c u l a t i o n output. The t i t l e 'Simplex I t e r a t i o n ' marked the be g i n n i n g of the i t e r a t i v e r o u t i n e , and j u s t -under the t i t l e were p r i n t e d the ranges over which the • search was to be performed (10,000 - 30,000 and 60,000 -9 0 , 0 0 0 ) . The FORTRAN program a u t o m a t i c a l l y generated Spendlay's matrix v a l u e s from the range v a l u e s and r e g r e s s e d the molecular weight p a i r s (see Equ a t i o n 5 2 ) . The r e s u l t i n g F - r a t i o s were presented as the f i r s t p a r t of the next r e g r e s s i o n , demarcated by a l i n e of s t a r s . A f t e r the f i r s t three r e g r e s s i o n s , the a c t u a l simplex o p e r a t i o n began, s i g n i f i e d by a p r i n t e d i t e r a t i o n v a l u e . The b e s t , worst, and next t o worst F - r a t i o s were then p r i n t e d . The i t e r a t i o n procedure then continued u n t i l i t f a i l e d due t o no improvement being o b t a i n e d i n the Figure 26, Flowchart for the simplex algorithm. Spendlay's matrix ** y N yes no Go to 1 Massive Reflection Contraction Fmcr > *r 2 End I Centroid (MCR) Go to 1 Centrdid (CR) Centroid F > F ' ? •re r Reflect Go to 1 Regression Regression Centroid (E) Reflection Contraction F r > F b ? expand Regression F > F ? e r F > F < F w r n F > F_ < P. n r b Centroid (R) Go to 1 F < F _ r w ? Worst Contraction Regression F > F ? cw w Centroid (CW) Centroid (MWC) F > F ? mwc w Regression Go to 1 Massive Worst Contraction Go to 1 Contraction \Failure - End C M 3. xx'>> > »">') >>>>>>< .. WARM._.. 1 . . Uif-b. ^ . 2 RI s !-.). i V j l i s ifrrrrr/>>t)>r ' r r x r X X xx x X x x x xx X xx XXXX XX.x ^« XXXX XXXX XXX x x x x x x x x x ~ ' " ~ " i> . -ARM. . .b . .%M.kH . . . 7. .WARM. . . 8. . WARM. .WARM . . 3 . ARM . . .4 . .W ARM , U M V E R S I TY CF K C L C M i u T I . ' .G ILMi-.tr' Ml S l / . R l i ' 7) - " . 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 . 9 . . WAR/-'. . . 0 . . W A R M . . . 1 . . W A R M . . . 2 . . W A F. M . • . 3 . i ' 5 : " l l " : 6 9 F R I APR 2 2 / 7 7 " ThE C C f F L T I N C J S I G rOMN C ENT FR U l L ' B . C I C S E C A L L C A r S A T . A P R I L 23 - R E N O V A T I O N S * * F i g u r e 2 7 . I n i t i a l o u t p u t o f t h e s i m p l e x " a l g o r i t h m . ' '" T T T T T T T T T T T T cctccccccccc I'M MM MM T M T T T T I T T T T C C C C C C C C C C C C ' " (•MM M MM M M M ' MMM TT cc cc »M<h M M V M yyyy MM M M TI cc CC yy yy yy ty yy yy MM MM T 1 cc ~ c c >M~ "MM MM"" ;•>• yy MMMM . MM T T cc cc yy yy ^ ^  MM MM MM. T T cc cc 1" M yy M M MM r r CC " cc CM t-y MM MM r r cc cc yy yy M.V MM TT cc cr. r M yy y y MM I r C C C C C C l ' C C ecu"" yy MM MM TT C C C C C C C C C C C C CM M M MM MM * * L A S T S I CNCN WAS: 15 : IC : 16 L S E R " T C H - " S I C N E C CN AT " 1 5 : 1 1 : 1 0 CN FR I APP 2 2 / 7 7 iXL.S C c i J . S S C C F A M - C e j . S I M l . C 2 1 b = — K 1JMA7 E A E C U T I C N b=C !NS M,MDER OF M C L E C U L A F W E I C F I S 2 M . . v ats. OF C A T A C ; ? C S 25 "NC. -EER " C - F ' C Y C C E S " 1 " " M C L c C U L A . ' . . J i C F T I M E R V J L 1 .7C C A L C U L A T I O N 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 TTER A T I CN MCC6 ' 2 » S M C O T H N I N G MODE r — - " - " t " " O ; The F e R T I N A N T P AR AM; TF. F S F C K T H I S C A I C I I A I I C K A F E : j • . ; C c L L D C T TCV 7 . 2 C C C O M F . M S C U S • 6 . 8 0 U O O ' ' J 121. co m -a* • m 'n o m p a * [O O r\i S i . a — a; o o • b o o !oo i ' 0 ; a: x > |< U Oi n a s J! - UV c* o u eg —• —» . n m —» <i m !~« o ••** Ul jy» n V* Ul r» o m vr> ut C» it Ul cj ut m , CM j"* m \* t/> r- m CM • I • c» • O j ci . o o O p O ci rn m I in m m •n en r<i 1 m m o n CJ o j CJ o o UJ UJ) Ul LU UJ 1 AJ UJ 1 UJ ) 1 IJJ 1 UJ 1 UJ ~« >n m u\ rM W r\J «># m rn '.-*> •o ' «-» —• •o o T> r* •o .o •o •a ru rg rg rg CN CN H sotsf t sstaa's •itit S9IDN3n33HJ 3 Hi -SI S3IH3S *H 3*1 S9S*tIS 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 -3«tf S3IDN3n03>H 3Wi •i8/9E V H V J -SI S3ld3S *W 3H1 '0<rL.\ 1X3M *8/9 EDS'U ? 1S31-3 . ..... » d33wriN * 3 U V _ l l l ISH3V '81*2 iS3B EDS* a ? 1S3J-J 6 I S » E ' I E&St 3*0 -3a? 5313N3nri3«d__i_ '8*583 • ?115 T -SI S3IS3S "W 341 *VO"3I*2 1S31-3 »**««»*«*»*»»*«»»««•**«**»*****•***•*»*****«*«*********»»**•«*•**»**********•****»**•*••***•**•«•*****»*.***********•*••**••** ' " " J L E i 9 f 3 - 3 H » S3I3N3n;>3_ 3-11 . ' i _?_L9. _ _ . ! _ ! _U__3.Ld__ l__Wi_ _ 355*>CL I 1S31-J _*.***tt*****4*********1>*.****.*.*.^**.**.*J*^ (-*»_•.•*-•.».•*>.«__•.*-• *•**.»*.•.•.•».•.»•.. V i E V c ' l e/ZZ9*3 -3M? S3IDN3nft3>H 3H1 -SI S3IW3S CH 3'U •00009 •0DD3I •00006 •00009 'ODOOE '00D3I CV ************************** 3NIinf)>l HTH1VM31I X33dHIS c 123. F - t e s t or due to the maximum l i m i t s e t on the i t e r a t i o n s b e i n g exceeded. The r e s u l t s of t h i s p a r t i c u l a r example were: M1 = 24,400 M 2 = 80,000 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 of a 25,000 - 80,000 d a l t o n mixture, p r e s e n t i n 1:1 r a t i o . F u r t h e r examples of the c a p a b i l i t i e s of the simplex o p t i m i z a t i o n r o u t i n e w i l l be presented l a t e r on. E. Smoothing of 'undefined data Although the i t e r a t i v e s o l u t i o n allowed the e s t i m a t i o n o f the c o r r e c t m o l e c u l a r weight, weight average m o l e c u l a r -weight, and the r a t i o s o f the i n d i v i d u a l components, t h e r e was one drawback. A complete MWD was not p r o v i d e d . The o r i g i n a l complete MWD o b t a i n e d a t one speed was usually-u n d e f i n e d and not very p r e s e n t a b l e . A r e p r e s e n t a t i v e d i s t r i b u t i o n c o u l d , however, be o b t a i n e d by combining the i n f o r m a t i o n from the i t e r a t i o n and the o r i g i n a l d i s t r i b u t i o n . T h i s was done by i n s e r t i n g a dummy statement i n t o the r e g r e s s i o n matrix to r e i n f o r c e t o c o r r e c t frequency v a l u e s , and thereby smoothing the d i s t r i b u t i o n . T h i s technique .93 f 2 = .92 124. was v e r y s i m i l a r t o t h a t used i n the removal o f the i n t e r c e p t from the r e g r e s s i o n . An example o f a dummy statement and where i t would appear i n the r e g r e s s i o n m a t r i x i s presented below. c ( 5 ) / c 0 l n K 2 n K 3n 4n 0.5 1.0 1.0 0.0 0.0 0.5 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0. •0.0 where a - end of the r e g r e s s i o n matrix b - smoothing s e t c - n u l l s e t T h i s dummy statement was f o r a two component case where the sum e q u a l l e d 1.0. The value s of 1.0 are a d d i t i o n a l dummy k^ _. v a l u e s which emphasize the m o l e c u l a r weights c o n t r i b u t i n g t o each component. The type o f dummy statement i l l u s t r a t e d c o u l d be entered i n t o the m a t r i x v i a a s p e c i a l s u b r o u t i n e u s i n g the r a t i o s o b t a i n e d from the i t e r a t i o n procedure. Performing a complete MWD c a l c u l a t i o n o f data i n c l u d i n g these dummy v a r i a b l e s l e d to the smoothing o f the o r i g i n a l undefined d i s t r i b u t i o n . Examples of the smoothing c a p a b i l i t y o f t h i s procedure can be seen i n F i g u r e s 2 8 and 29, the smoothed v e r s i o n s of F i g u r e s 19 and 20. 125. 1.0 0.6 f 0.2 14 28 56 M x 10 -3 112 224 Figure 28. Semilcgarittimic plot of f (M) vs. M for a 25,000-80,000 dalton irdxture (1:1 r a t i o ) . Single speed data; interval -- 2.0, with data having undergone smoothing. 0.3 0.2 Y 0.1 0.0 640 M x 10 ?igure 29. SemllcjgafltiTmic plot of f (M) vs. M for a 25,000-80,000-320,000 dalton mixture (1:1:1 r a t i o ) . Single speed data; interval =2.0, with data having undergone siroDtljing. 1 2 6 . Case Studies of Some Model Systems In order to i l l u s t r a t e the po t e n t i a l of the MWD ca l c u l a t i o n with respect to protein systems, four case studies of proteins w i l l be presented. The chosen systems represented situations that could ari s e during an u l t r a -c e n t r i f u g a l investigation of proteins. The systems included a discrete s e l f - a s s o c i a t i n g system, a three component rvxture, and two contrasting i n t e r a c t i n g systems. A l l were generated using the Rinde equation model and had the same p a r t i a l s p e c i f i c volume and extinctio n c o e f f i c i e n t . The models were analysed using multiple speed data (three speeds) and unispeed data. The weight average molecular weights of the components were calculated for some of the d i s t r i b u t i o n s using Equation 5 1 . The simplex i t e r a t i v e routine was used to obtain the best f i t weight average molecular weight, and hence the component concentration, i n order for comparison to the o r i g i n a l model. A. Catalase ( 5 7 , 0 0 0 - 2 3 2 , 0 0 0 daltons; 1:2 ratio) Catalase i s known to self-associate to form a tetramer. This type of int e r a c t i o n i s common to proteins and i s often affected by environmental factors, such as 127. pH and i o n i c s t r e n g t h . The MWD c a l c u l a t i o n c o u l d b e u s e f u l i n d e t e c t i n g t h e e x t e n t o f i n t e r a c t i o n . I n t h i s c a s e s t u d y , t h e t e t r a m e r was p r e s e n t i n t w i c e t h e c o n c e n t r a t i o n o f t h e monomer. A n a l y s i s o f t h e m u l t i s p e e d d a t a p r o d u c e d t h e d i s t r i b u t i o n p r e s e n t e d i n F i g u r e 30. A s s e s s m e n t o f t h e m o l e c u l a r w e i g h t f r o m t h e d i s t r i b u t i o n i t s e l f , u s i n g E q u a t i o n 51, p r o d u c e d t h e f o l l o w i n g r e s u l t s : M1 = 55,800 f± = .25 M 2 = .253,000 f = .75 T h e s e r e s u l t s were n o t u n r e a s o n a b l e , b u t were n o t v e r y a c c u r a t e when compared t o t h e o r i g i n a l m o d e l . T h i s was p a r t i a l l y due t o t h e d i s t r i b u t i o n b e i n g somewhat u n d e f i n e d . U s i n g t h e same d a t a , t h e d i s t r i b u t i o n c o u l d be s u b j e c t e d t o t h e s m o o t h i n g r o u t i n e , a f t e r w h i c h i t c h a n g e d t o t h e much more d e s i r a b l e f o r m p r e s e n t e d i n F i g u r e 31. C a l c u l a t i o n o f t h e w e i g h t a v e r a g e m o l e c u l a r w e i g h t u s i n g E q u a t i o n 51. p r o d u c e d : M± = 61,600 M 2 = 231,000 The a p p l i c a t i o n o f t h e s i m p l e x i t e r a t i o n r o u t i n e t o t h e o r i g i n a l d a t a p r o d u c e d t h e f o l l o w i n g v a l u e s : f± = .31 f 2 = .68 128. O-0 / \ o , ' o \ o , O \ / V i\ 1 O 0-J I I \ 15 30 60 120 240 480 960 M x 10~ 3 Fig\ore 30. Semlogaritlimic p l o t o f f (M) vs. M f o r a 57,000-232,000 dalton mixture (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 Figure 31. SeTmlcigaritljmic p l o t o f f (M) v s . M f o r a 57,000-232,000 dalton mixture (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, with data having undergone smootliing. = 57,000 = 232,000 f ± = .33 f 2 = .66 These res u l t s were i n excellent agreement with the o r i g i n a l model. Analysis of single speed data produced the d i s t r i b u t i o n i l l u s t r a t e d i n Figure 32, a better d i s t r i b u t i o n than for the o r i g i n a l three speed data. This s i t u a t i o n arose i n some cases, i l l u s t r a t i n g how the choice of speed could 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 . C a l c u l a t i o n of the weight average molecular weight using Equation 51 produced: f ± = .28 f 2 = .72 Applying the simplex optimization routine to unispeed data, produced the following: M1 = 52,300 M 2 = 237,000 M 1 ..= 56,700 M2 = 232,000 f 1 = .33 f 2 = .66 Again t h i s was i n excellent agreement with the o r i g i n a l model. 130. B. Trypsin i n h i b i t o r - ovalbumin - conalbumin (17,000-45,000-85,000; 1:1:1 ratio) This case represents a mixture of three proteins r e l a t i v e l y close together i n molecular weight with no interactions taking place. The f i r s t analysis of three speed data was made using a molecular weight i n t e r v a l of 2.0, with the r e s u l t i n g d i s t r i b u t i o n presented i n Figure 33. Cnly two components could be detected, mainly due to the poor resolution r e s u l t i n g from the r e l a t i v e l y large i n t e r v a l . By changing the i n t e r v a l to 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 Figure 3.4. Three components were then evident, although the d i s t r i b u t i o n became somewhat undefined. Further manipulation of the i n t e r v a l and range would enable a defined three component d i s t r i b u t i o n to be obtained. This operation was not required since the simplex optimization could provide the solution. The r e s u l t s of the simplex optimization were: M 1 = 16,800 M 2 = 43,000 M 3 = 84,000 Single speed simplex optimization produced: f x = .32 f„ = .32 f 3 = .35 131. 0.8 0.6 O * S 0.4 0.2 O-o o o 0.0 15 30 60 _G—1_ 120 240 480 M x 10 -3 960 Figure 32. SeMlogari+JnTtdc plot of f (M) vs. M for a 57,000-232,000 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 17,000-45,000-85,000 dalton mixture TT:1:1 ratio). 10,000, 20,000 and 30,000 rpm, interval = 2.0. 132. M 1 = 16,400 M2 = 42,200 M = 84,600 3 Figure 35 shows a smoothed function using the r e s u l t s obtained from the single speed i t e r a t i o n . The smoothing routine did not, i n t h i s case, produce exactly the d i s t r i b u t i o n desired, p a r t i a l l y due to the empirical nature of the c a l c u l a t i o n . The smoothing routine required that the region of best f i t frequency response be strengthened, which was not always possible when the molecular weights were so c l o s e l y spaced. C. . a s^ -casein and K -casein, and interaction, product (25,000-400,000-1,500,000; 2:2:1 ratio) This i n t e r a c t i n g system represents one of the more d i f f i c u l t systems studied. Both °t.s^ and K-casein are low molecular weight components, but K-casein aggregates to a very high molecular weight. When as^-casein was introduced into the system, an in t e r a c t i o n product of intermediate molecular weight was formed. Analysis of t h i s system at three speeds produced the d i s t r i b u t i o n i l l u s t r a t e d i n Figure 36. In the c a l c u l a t i o n s , the molecular weight i n t e r v a l had been increased to 2.2, to f, = .31 133. n.50 & 0.25! o 0.00 o -Cl-10 Figure 34. i \ 22 O O O o n t \ j 50 M x 10 -3 o \ o \ Q O 113 SemilogaritJmLc p l o t o f f (M) vs. M f o r a 17,000-45,000-85,000 dalton mixture (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 s ^ 0.25 O.. o o o - O o J o o o o / o o 0.00 o o \ - O L 10 Figure 35. 22 50 110 M x 10 -3 Seiiiilogarithmic p l o t o f f (M) v s. M f o r a 17,000-45,000-85,000 dalton mixture (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. 134. accommodate the wide range of molecular weights. By applying the simplex optimization routine to multiple speed data, the following r e s u l t s were obtained: M x = 24,000 M 2 = 399*, 000 M3 = 1,491,000 The d i s t r i b u t i o n obtained from a single speed can be seen i n Figure 37. Simplex i t e r a t i o n for the single speed data produced: M1 = 24,000 f 1 = .39 M 2 = 402,000 f 2 = .39 M 3 = 1,550,000 f 3 = .21 D. Lysozyme - ovalbumin and i n t e r a c t i o n product (17,000-45,000-62,000; 2:2:1 ratio) This study was of an in t e r a c t i o n at the other extreme of the resolution scale. In t h i s case, the i n t e r a c t i o n product d i f f e r e d l i t t l e from the molecular weight of ovalbumin. Analysis of multispeed data produced a two peak d i s t r i b u t i o n . Simplex optimization for three components generated the following r e s u l t s : 1 f 1 = .39 f 2 = .39 f 3 = .19 135. M = 17,000 f± = -40 M 2 = 45,800 f 2 = .41 M0 = 63,000 ' f , = -18 3 J Analysis of single speed data using the simplex optimization produced the following r e s u l t s : Mx = 16,900 f x = -39 M_ •= 42,000 f ? = .31 M 3 = 59,300 f 3 = -29 In t h i s case, the simplex c a l c u l a t i o n was not able to y i e l d r e s u l t s as accurate as i n previous systems. This could have been a speed related problem, where the single speed d i d not carry enough information to locate the correct optimum. Discussion In a l l cases, the three speed data produced f a i r l y reasonable d i s t r i b u t i o n s . A l l of these d i s t r i b u t i o n s provided an in d i c a t i o n of the number of components present, and of the p a r t i c u l a r molecular weight regions i n which the simplex would begin i t s search. The simplex optimization worked very well, with a l l multispeed cases giving accurate re s u l t s that r e f l e c t e d the r a t i o s and molecular weights of the o r i g i n a l models. From t h i s 136. investigation i t was apparent that i f the c e n t r i f u g a l data approached the accuracy of the model data, the simplex optimization procedure could r e a d i l y solve for r e l a t i v e concentrations, i n a wide var i e t y of two and three component systems. In the case of single speed data almost the same conclusions could be made, except i n the rather d i f f i c u l t case of the lysozyme-ovalbumin in t e r a c t i o n . . Use of the simplex optimization removed to some extent the require-ment that a defined d i s t r i b u t i o n be obtained, as long as a good i n d i c a t i o n was provided as to the number of components present i n a given system. This i n d i c a t i o n could come from either the MWD or from previous knowledge of the components i n the mixture. Analysis of Protein Mixtures Up to t h i s point only model systems have been analysed. Following the development and v e r i f i c a t i o n of the computer programs, and the f i n a l removal of a l l the problems, some ultra c e n t r i f u g e runs were completed. These runs were made with standard proteins dissolved i n 0.05 M phosphate buffer at pH 7, dialysed for 24 hours, d i l u t e d to an absorbance of 0.5 and then mixed i n a 1:1 r a t i o by volume. The experimental procedures described for u l t r a c e n t r i f u g a t i o n in Part I were carr i e d out for these runs. 137. 0.4 M x 10 Figure 36. SerMlogarithmic p l o t o f f (M) vs. M f o r a 25,000-400,000-1,500,000 dalton mixture (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 0.3 S 0.2 0.0 O / i / 1 / i O / \ ' G G \ / ° G > I , \ / \ -/ 1 o o 1 i O 1 O / V \ o /' ' i O i 0 » 1 \ l » 1 I 1 ' I / 1 1 » I I 1 1 v i n i in i ; Q 1 n\ i rt> i l i \ O i 0\ 6 29 140 680 3292 M x 10 3 Figure 37. Serrdlogarithmic p l o t of f (M) vs. M f o r a 25,000-400,000-1,500,000 dalton mixture (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 with the centrifuge during our attempts to obtain data for protein mixtures. The d i f f i c u l t y i n obtaining a correct baseline was one of the most serious problems. What e f f e c t an incorrect baseline could have on the MWD i s unknown. Differences i n p a r t i a l s p e c i f i c volume and e x t i n c t i o n c o e f f i c i e n t s among proteins caused further complications i n experimenting with r e a l systems. Since the program could 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 value was used. The following experimental re s u l t s represent some of the more successful runs. A l l the samples were run at multiple speeds with the analyses being performed mostly on three speed data. A. Trypsin inh ib1tor-conalbumin (1:1 by volume) The r e s u l t s of t h i s experiment were very encouraging. The mixture was run at three speeds: 9,500, 14,400 and 19,600 rpm. The C ( ^ ) / C q data were run through the complete MWD 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 seen i n Figure 38. A well defined d i s t r i b u t i o n was obtained with the weight average molecular weights determined by Eguation 51 as follows: 139. M 1 = 23,000 f 1 = .52 M 2 = 82,000 f 2 = .49 The d a t a were t h e n p u t t h r o u g h t h e s m o o t h i n g r o u t i n e t o p r o d u c e t h e d i s t r i b u t i o n i l l u s t r a t e d i n F i g u r e 39. When t h e raw d a t a were p u t t h r o u g h t h e s i m p l e x o p t i m i z a t i o n r o u t i n e , t h e b e s t f i t s o l u t i o n was f o u n d t o b e : M 1 = ?4,000 f± = -51 M 2 = 98,000 f 2 = .47 A n a l y s i s o f d a t a o b t a i n e d o n l y f r o m a s i n g l e s p e e d (19,620 rpm) p r o d u c e d t h e d i s t r i b u t i o n p r e s e n t e d i n F i g u r e 40. A n a l y s i s o f t h e s e d a t a , u s i n g t h e s i m p l e x o p t i m i z a t i o n r o u t i n e , p r o d u c e d : M± = 12,700 f1 = .28 M 2 = 64,800 f 2 = .63 B. ' T r y p s i n i n h i b i t o r - o v a l b u m i h - c o n a l b u m i n (1:1:1 b y volume) A n a l y s i s o f t h r e e s p e e d d a t a (9,600, 15,500 and 21,600 rpm) p r o d u c e d t h e d i s t r i b u t i o n i l l u s t r a t e d i n F i g u r e 41. S i m p l e x o p t i m i z a t i o n o f t h i s d a t a s e t f o r t h r e e components g e n e r a t e d t h e f o l l o w i n g optimum v a l u e s : 1 4 0 . a. 8 a.6 r S 0 . 4 0 . 2 0 . 0 14 28 56 112 M x 10 224 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 21,600 rpm. 0 . 8 JO, 6 g 0 . 4 3-2 u-0 ' o o o Q 14 28 56 112 224 M x 10 - 3 Figure 39 . 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 0.6 S 0.4 0.2 0.0 o - 0 € > o 1 \ 0 1 \ / \ o , 1 \ 1 1 \ 1 1 / \ - o o \ 1 \ 1 0 \ / / O i \ 1 O I O I \ 1 14 28 56 M x IO - 3 112 224 Figure 40. Serailogarithmic plot of f (M) vs. M for a mixture of trypsin inhibitor and conaJ^urrdn. Absorbance =0.5 for both, mixed 1:1 by volume. Run at 19,600 rpm. 0.50 LLT 0.25 O „ 0 1 i o' x I' \ 1 , ? i / \ ' o i \ / O \ I 1 O i n i \ i n l 1 r> 7 14 28 56 112 224 -3 M x 10 Figure 41. SemLlcigarittimic plot of f (M) ' vs. M for a mixture of trypsin inhibitor, ovalbumin and conalburtiin. Absorbance = 0.5 for a l l , iriixed 1:1:1 by volume. Run at 9,600, 15,500 and 21,600 rpm. 14 2. M± = 10,000 f± = .37 M 2 = 49,128 f 2 = .43 M3 = 115,000 f 3 = .22 C. Ovalbumin-thyroglobulin (1:1 by volume) This system i n r e a l i t y represents a three component d i s t r i b u t i o n since thyroglobulin i s made up of two subunits. The d i s t r i b u t i o n obtained from three speeds (6,503, 9,620 and 15,526 rpm) i s presented i n Figure 42, which shows the presence of three components. Simplex i t e r a t i o n of the data produced the following r e s u l t s : Mx = 37,000 ± 1 = .43 M 2 = 388,000 f 2 = .17 M3 = 1,040,000 f 3 = .21 D. Catalase Catalase i s a two component system since i t e x i s t s i n two forms, a monomer and a tetramer. This was demonstrated by the d i s t r i b u t i o n , even though i t was undefined, obtained at three speeds and presented i n Figure 43. Simplex i t e r a t i o n of the data produced: 143. 0.50 G / \ / G G O l \ \ o o i • > o N 7 o G \ G O o o' * —LO 1 Ci 1 1 , l_ o_ 20 80 320 1280 M x 10~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 at 6,300/ 9,600 and 15,500 rpm. M x 10 3 320 1280 Figure 4.3. Semilcgarittimic plot of f (M) vs. M for catalase. Absorbance = 0.5, run at 9,600,15,500 and 21,600. 144. M M a = 54,800 - = 379,000 40 60 E. D i s c u s s i o n A l t h o u g h t h e e x p e r i m e n t s performed were n o t as c o n c l u s i v e as d e s i r e d , t h e y d i d i l l u s t r a t e t h a t t h e MWD 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 from t h e system. Even though t h e m o l e c u l a r w e i g h t s and 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 m o l e c u l a r w e i g h t s and r a t i o s t h a t were w i t h i n r e a s o n a b l e c o m p a r i s o n w i t h t h e i r known v a l u e s . I n o r d e r f o r e x p e r i m e n t a l 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 d a t a w ould 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 two o r t h r e e speeds. S i n g l e speed d a t a has n o t been g i v e n any prominence i n t h e a n a l y s i s o f t h e e x p e r i m e n t s . T h i s was m a i n l y due 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 speed and t h r e e speed r e s u l t s , as shown i n t h e c a s e o f t r y p s i n i n h i b i t o r - c o n a l b u m i n . 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 speeds chosen 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 . C o n s i d e r i n g t h e known c o m p o s i t i o n o f t h e systems, t h e m u l t i s p e e d d a t a were more r e p r e s e n t a t i v e t h a n most o f t h e s i n g l e speed r e s u l t s . 145. A further factor to be considered was the u l t r a -centrifuge i t s e l f . Although the preparative u l t r a c e n t r i f u g e was useful for the routine assessment of molecular weights, i t was questionable whether the data obtained were s u f f i c i e n t l y accurate for the determination of molecular weight d i s t r i b u t i o n s . Discussions with other workers (40) revealed some concerns whether the preparative u l t r a -centrifuge could be used for t h i s r e l a t i v e l y sophisticated analysis. In addition, UV optics lacked "the inherent pr e c i s i o n of schlieren or RayleLgh interference opt i c s " (12). Obviously, the best method for the assessment of experimental data would be the use of a Model E ult r a c e n t r i f u g e equipped with Raleigh interference o p t i c s . Evaluating the c a p a b i l i t i e s of the MWD c a l c u l a t i o n using a less sophisticated instrument may leave some doubt as to the v a l i d i t y of the r e s u l t s . Therefore, the experiments performed can only be considered to provide an i n d i c a t i o n that the MWD of proteins could be obtained. Evaluation of the actual c a p a b i l i t i e s of the c a l c u l a t i o n w i l l have to be l e f t to groups having access to a Model E ultracentrifuge and a d e t a i l e d knowledge of the protein systems to be assessed. Other Factors A. E x t i n c t i o n c o e f f i c i e n t s 146. The derivation of the Rinde equation assumed equivalent e x t i n c t i o n c o e f f i c i e n t s so that the equation could be written i n terms of concentration without ambiguity. I t i s well known that t h i s i s not the case, since extinction c o e f f i c i e n t s of proteins can vary markedly. To account for varied e x t i n c t i o n c o e f f i c i e n t s , the formulation had to be changed. Absorbance i s related to concentration by 3eer's law: A = abc where: A - absorbance a - abso r p t i v i t y b - length of the l i g h t path (cm) c -.concentration (g/1) Reformulating the Rinde equation i n these terms gives: A ( £ ) \ - > X.M. exp ( - X.M . C ) = L L — f i E<2- 6 4 A_ 1 - exp(-A .M. ) 1 exp(-A.M.) where; A . a.c . f = Q 1 _ 1 o i A_ A o o Eq. 65 14 7. Expanding the summation and reducing the bulk term to k produces: A (£) a . c . a . c . a c . = -=-°i K, + -=-°i K„ . . . + - 3 - S a K Eq. 6 6 A A A A o o o o By examining the above equation i t can be seen that the contribution of each component i s d i r e c t l y related.to i t s e x t i n c t i o n c o e f f i c i e n t . A component with twice the absor p t i v i t y of another would act as i f i t was present i n twice the amount, and therefore have twice the area i n the calculated MWD. In the case of well resolved d i s t r i b u t i o n s of di s c r e t e protein components, a simple area adjustment should s u f f i c e to correct the di f f e r e n c e . This was shown to be the case when equilibrium patterns of two proteins, having d i f f e r e n t molecular weights and a b s o r p t i v i t i e s , were synthesized, 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 constant 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 could be corrected. This correction was only useful when the components were completely separated, or the regions of t h e i r d i s t r i b u t i o n s well defined. B. P a r t i a l s p e c i f i c volume Differences i n the p a r t i a l s p e c i f i c volume of proteins have been ignored up to t h i s point i n the d i s s e r t a t i o n . 148. Any d i f f e r e n c e i n t h e p a r t i a l s p e c i f i c v o lume b e t w e e n d i f f e r e n t components m a k i n g up a s y s t e m , a f f e c t e d t h e MWD. I n t h e p r o g r a m d e v e l o p e d b y S c h o l t e , a n d i n t h e m u l t i p l e r e g r e s s i o n a p p r o a c h u t i l i z e d i n t h i s t h e s i s , 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 v o l u m e o f t h e components c o u l d n o t be a c c o u n t e d f o r , and t h e u s e o f a mean v a l u e was r e q u i r e d . However, t h i s n e e d n o t be t h e c a s e . The p a r t i a l s p e c i f i c volume o f e a c h component c o u l d be a s s i g n e d t o a p o r t i o n o f t h e r a n g e u s e d i n t h e MWD c a l c u l a t i o n . When t h a t r e g i o n o f t h e r a n g e was u s e d f o r t h e c a l c u l a t i o n o f t h e K_^ _. t e r m o f t h e r e g r e s s i o n m a t r i x , t h e c o r r e c t p a r t i a l s p e c i f i c v o lume w o u l d b e i n e f f e c t . T h i s c o n c e p t c a n b e s t be e x p l a i n e d b y a n e x a m p l e : a p r o t e i n o f 50,000 d a l t o n s s e l f - a s s o c i a t e d t o f o r m a t e t r a m e r t h a t had a d i f f e r e n t p a r t i a l s p e c i f i c v o l u m e t h a n t h e o r i g i n a l . ' By r e l a t i n g t h e p a r t i a l s p e c i f i c v o lume o f t h e monomer t o a m o l e c u l a r w e i g h t r e g i o n o f 10,000 t o 80,000, and t h e p a r t i a l s p e c i f i c volume o f t h e t e t r a m e r t o the. h i g h e r v a l u e s o f t h e r a n g e , a l l t h e K_^ _. t e r m s i n t h e r e g r e s s i o n m a t r i x c o u l d be c a l c u l a t e d u s i n g t h e c o r r e c t p a r t i a l s p e c i f i c volume f o r e a c h component. T h i s a p p r o a c h was u s e d i n t h e o r i g i n a l Monroe 1880 p r o g r a m and was f o u n d t o work w e l l . I t must be e m p h a s i z e d , however, t h a t p r o p e r u s e o f t h i s p r o c e d u r e r e q u i r e s some p r i o r k n o w l e d g e o f t h e s y s t e m u n d e r s t u d y , s o t h a t t h e 149. p a r t i a l s p e c i f i c volumes could be assigned to the correct molecular weight region. The above procedure was not included i n the FORTRAN program since the i n i t i a l objective was to compare the r e s u l t s with Scholte's l i n e a r programming program that used lambda as a precalculated input term. In order to use the above approach, lambda had to be calculated as part of the program. 150. CONCLUSION The i n i t i a l objective of automating the u l t r a -centrifuge f o r the purpose of obtaining molecular weights for both homogeneous and heterogeneous systems was t e c h n i c a l l y successful. With t h i s system, routine analysis of proteins could be made with ease and r e l a t i v e l y l i t t l e tedium. However, the problem of correct baseline determination when using UV optics s t i l l has to be eliminated to make the system more useful. Other workers (7,46) have recognized the s i g n i f i c a n c e of the baseline problems, and a solution for determining the true baseline was suggested by Chernyak and Magretova (6). This correction was investigated and shown to be useful for homogeneous systems (.34), but not applicable to heterogeneous systems. U n t i l the baseline problem i s overcome, the accuracy of molecular weight determinations using the UV scanner i s open to question. The use of desktop computing f a c i l i t i e s allowed the analysis of the u l t r a c e n t r i f u g a l data to be performed quickly and accurately, using regression methods that were out of reach of many smaller laboratories only a few years ago. These curve f i t t i n g techniques are now r a p i d l y becoming the standard method of analysing a l l types of 151. experimental data (16). The programmable f e a t u r e s on the desktop c a l c u l a t o r a l l o w anyone w i t h a minimum of mathematical knowledge t o program complex mathematical f u n c t i o n s , making them r e a d i l y a v a i l a b l e f o r r e p e t i t i v e use. The combination of the a c q u i s i t i o n system and desktop c a l c u l a t o r f a c i l i t a t e d the s u c c e s s f u l p r o c e s s i n g of experimental u l t r a c e n t r i f u g a l data i n t o the d e s i r e d f i n a l r e s u l t s . The second o b j e c t i v e of u s i n g MWD c a l c u l a t i o n s f o r the study of p r o t e i n s became of minor importance i n comparison w i t h the i n v e s t i g a t i o n o f m u l t i p l e r e g r e s s i o n as an a l t e r n a t i v e method of o b t a i n i n g a d i s t r i b u t i o n . T h i s study was launched when d r i v e problems prevented use of the c e n t r i f u g e f o r an extended p e r i o d of time. Although the approach to s o l v i n g the MWD c a l c u l a t i o n f o r p o s i t i v e frequency v a l u e s was somewhat e m p i r i c a l , the r e s u l t s bore out the assumption t h a t any 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 n e g a t i v e frequency v a l u e s were not p a r t of the s o l u t i o n . The l i n e a r programming method of S c h o l t e made no such assumptions, but came to the same c o n c l u s i o n . The s t a t i s t i c a l parameters were not found t o be as u s e f u l as thought by Magar (31), but were s u c c e s s f u l l y implemented as a v e h i c l e f o r an i t e r a t i v e s o l u t i o n . The major drawback to the method o f S c h o l t e was the numerous speeds necessary f o r the MWD to be 152. resolved completely. This was a genuine problem since a seven speed run, that required equilibrium to be reached at each speed, was extremely time consuming and demanding on the instrument. Furthermore, the accuracy of the data was d i l u t e d considerably by the repeated measurements that required redefining of the meniscus and c e l l bottom at each speed. The use of 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 frequencies of the i n d i v i d u a l components to be elucidated frrm the data of a single speed run. One assumption was necessarily 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 to form d i s t i n c t peaks. I f that was not possible, estimates of the molecular weights should be known. The re s o l u t i o n of the i t e r a t i v e method was shown to be much greater than that of the complete molecular weight.distribution i t s e l f . In general, the d i s t r i b u t i o n c a l c u l a t i o n required that a factor of two separated the molecular weight of the components. However, a s i g n i f i c a n t l y smaller factor was allowable for the i t e r a t i o n i f an estimate of the component molecular weights was ava i l a b l e . This was i l l u s t r a t e d i n the case of the model i n t e r a c t i o n of lysozyme and ovalbumin. Undefined d i s t r i b u t i o n s were generally obtained from single speed data and a smoothing routine was developed to convert these data into a more suit a b l e 153. form. 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 and 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 r e p r e s e n t e d . The a c t u a l a p p l i c a t i o n of the complete MWD c a l c u l a t i o n t o r e a l p r o t e i n mixtures was not f u l l y i n v e s t i g a t e d i n t h i s work. In order t o t r u l y e v a l u a t e the p o t e n t i a l accuracy of the method, experiments should be performed 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 mixtures, u s i n g an instrument of known c a p a b i l i t y and accuracy. Since 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 an unknown q u a n t i t y i n terms of o v e r a l l accuracy and r e l i a b i l i t y , r e a l problems were posed i n u s i n g t h i s instrument to assess such a r e l a t i v e l y s o p h i s t i c a t e d t e c h n i q u e . Workers i n the area of u l t r a c e n t r i f u g a t i o n c o n s i d e r the Model E u l t r a c e n t r i f u g e to be the minimum standard f o r any a n a l y t i c a l work. In r e s p e c t t o the equipment a v a i l a b l e , the e v a l u a t i o n of the MWD c a l c u l a t i o n became a s e l f d e f e a t i n g e f f o r t . However, the few experiments performed seemed to produce the fundamental i n f o r m a t i o n r e q u i r e d to o b t a i n a d i s t r i b u t i o n , and i n d i c a t e d t h a t the r e s u l t s were b a s i c a l l y i n l i n e w i t h the known composition of the sample. As mentioned p r e v i o u s l y , an estimate of the molecular weights or the range of molecular weights p r e s e n t i n the 154. sample under i n v e s t i g a t i o n would be u s e f u l i n o b t a i n i n g the MWD. There were a number of techniques and methods a v a i l a b l e t h a t c o u l d h e l p e s t a b l i s h these parameters f o r p r o t e i n systems. The most obvious method was to run the i n d i v i d u a l components s i m u l t a n e o u s l y w i t h the mixture i t s e l f , i n a m u l t i p l a c e r o t o r , u s i n g e i t h e r c o n v e n t i o n a l c e n t r i f u g e or Yphantis c e l l s . T h i s allowed the de-t e r m i n a t i o n , under i d e n t i c a l c o n d i t i o n s , of the molecular weights of the i n d i v i d u a l components and of the component mixture. Another u s e f u l approach was to use curve f i t t i n g t echniques such as m u l t i p l e r e g r e s s i o n o r o r t h o g o n a l p o l y n o m i a l s t o smooth the raw data (11,-38) . I f a r e a s o n a b l e f i t c o u l d be made, the m o l e c u l a r weights along the s o l u t i o n column c o u l d be o b t a i n e d , i n t u r n p r o v i d i n g an estimate of the range f o r the MWD c a l c u l a t i o n . A s i n g l e p o l y n o m i a l might not s u f f i c e i n a l l cases and s e v e r a l c o u l d be r e q u i r e d to o b t a i n a good f i t of the complete 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 (26). Other techniques t h a t c o u l d be u s e f u l are the approach t o e q u i l i b r i u m method of A r c h i b a l d (17) t h a t allows an estimate of the h i g h molecular weight component to be o b t a i n e d , and the Yphantis method (68) t h a t c o u l d y i e l d an e s t i m a t e of the low m o l e c u l a r weight component. One other s e r v i c e a b l e method i s the 'two s p e c i e s p l o t ' a n a l y s i s used by Yphantis and Roark (69), l a t e r a p p l i e d to p r o t e i n 155. mixtures by J e f f e r y and Pont (25). The use of some o f these a n c i l l a r y methods c o u l d h e l p to c h a r a c t e r i z e the system and e s t a b l i s h whether the r e s u l t s of the MWD c a l c u l a t i o n and the simplex i t e r a t i o n were reasonably c o r r e c t . Many workers f o r over f o r t y years have attempted to o b t a i n a c c u r a t e MWDs from u l t r a c e n t r i f u g a l d a t a . The computer has allowed new methods of s o l u t i o n t o o l d problems, as i n the case of the Rinde equation. However, the computational s o p h i s t i c a t i o n of the methods used t o o b t a i n a MWD from the Rinde equation does not guarantee t h a t the problem i s s o l v e d . T h i s i s because i t has been r e c o g n i z e d of l a t e t h a t the Rinde equation i t s e l f i s the major d e t e r r e n t to o b t a i n i n g a p e r f e c t MWD, s i n c e the Rinde equation i s an Improperly Posed Problem. T h i s means t h a t there are an i n f i n i t e number of s o l u t i o n s t o the e q u a t i o n . In r e a l i t y , however, onl y a l i m i t e d number of s o l u t i o n s w i l l e x p l a i n the d a t a . The methods of Donnelly, S c h o l t e and Gehatia and W i f f attempt to f i n d these l i m i t e d s o l u t i o n s . T h i s i s a l s o t r u e of the m u l t i p l e r e g r e s s i o n -simplex approach. From the study of model data i t has been. shown t h a t the method of S c h o l t e and the m u l t i p l e r e -g r e s s i o n approach work e q u a l l y w e l l . The m u l t i p l e r e g r e s s i o n s o l u t i o n , however, has the advantage of a l l o w i n g the s t a t i s t i c a l parameters a v a i l a b l e to be used as a guide to s o l v i n g f o r the b e s t f i t f r e q u e n c i e s and m o l e c u l a r 156. weights. T h i s advantage i s s u b s t a n t i a l , s i n c e i t t h e o r e t i c a l l y a l l o w s the e v a l u a t i o n o f the m o l e c u l a r weights and c o n c e n t r a t i o n o f the components o f a mixture a t one r o t o r speed. Whether t h i s i s p o s s i b l e i n r e a l i t y i s unknown, but c e r t a i n l y fewer r o t o r speeds w i l l be r e q u i r e d i f the m u l t i p l e r e g r e s s i o n approach i s used. 157. LITERATURE CITED Adams, E.T. J r . , W.E. Ferguson, P.J. Wan, J.D. Sarquis, B.M. Escot. 1975. 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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 . B i o c h e m i s t r y 10:3241. 164. APPENDIX A. Proteins Used i n t h i s Investigation Protein Bovine serum albumin Catalase Conalbumin Lysozyme Ovalbumin Thyroglobulin Trypsin Inhibitor Manufacturer Sigma Sigma Sigma Worthington ICN Pharmaceuticals Sigma Calbiochem B. Assembly of the Yphantis C e l l s The Yphantis centrifuge c e l l required s p e c i a l handling, since i t d i d not have any f i l l i n g holes and tended to leak i f the following procedure was not followed. When assembling the Yphantis c e l l , two window gaskets and two screw ring gaskets should be used rather than one. The c e l l was assembled empty, the screw ring torqued to 110 lb.-in.(916 newton-centimeters), and the f i l l i n g hole screws put into place against the centerpiece w a l l . The screw r i n g was then released and the top window and holder removed by l i g h t tapping against a s o l i d surface. The c e l l sectors could then be f i l l e d with a syringe. After 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 r e i n s e r t e d and the screw r i n g torqued to 110 l b . - i n . The f i l l i n g h o l e screws were then r e l e a s e d , the screw r i n g r e t o r q u e d to 135 lb.-in.( 1125 newton-centimeters) and the f i l l i n g h o l e screws r e t i g h t e n e d . I f t h i s procedure was f o l l o w e d , the c e l l c o u l d be run a t h i g h speeds without i n c i d e n c e of leakage. C. Programs f o r the Monroe C a l c u l a t o r A b r i e f d e s c r i p t i o n of t h r e e programs mentioned i n the t e x t o f t h i s work w i l l be p r o v i d e d below, along w i t h a note of t h e i r use. 1. Time t o reach e q u i l i b r i u m program d e r i v e d by Van Holde and Baldwin (54) i n o r d e r t o p r o v i d e an estimate of the time r e q u i r e d f o r a system to r e a c h e q u i l i b r i u m . The r e l a t i o n i s : T h i s program was w r i t t e n to s o l v e the r e l a t i o n t e (b - m) 2 F(a)/D Eq. IA where: U(a) = 1 + ( l / 4 ^ 2 a 2 ) Eq. 2A a = RT/M(1 - v p ) r (b - m) Eq. 3A Eq. 4A F i g u r e IA. Time to reach e q u i l i b r i u m program Input: A - d i f f u s i o n c o e f f i c i e n t - e - molecular weight - 1 - v p - rpm - r Output: B - time i n hours 167 5 « JO.0 0 0 0 0 0 0 - 0 1 1 < . 0 0 0 0 0 0 0 0 0 - 0 3 a -• 6 1 8 0 0 0 0 0 0 0 4 2 « . 6 8 0 0 0 0 0 0 0 - 0 1 1 < . 2 0 0 0 0 0 0 0 0 C 4 7 . 0 0 0 0 0 0 0 0 0 0 0 4 . 0 0 0 0 3 0 0 0 0 - 0 1 2 • 4 3 7 3 4 5 8 0 0 } 0 4 - 0 5 9 7 0 0 5 0 3 0 1 A 168. 2 -1 -7 d i f f u s i o n coeffxcient (F.icks - cm sec x 10 ) c e l l bottom meniscus (b + m)/2 measurement of the departure from e q u i l i b r i u m By knowing the length of the solution column and the estimated molecular weight and d i f f u s i o n c o e f f i c i e n t , the time to reach equilibrium could be determined. The error term £ r e f e r s to the deviation of the pattern from i t s true equilibrium position and was usually set at 0.001. The input and output of t h i s program are presented i n Figure IA. 2. Orthogonal polynomial curve f i t t i n g program This was an alternative curve f i t t i n g program to multiple regression. This curve f i t t i n g technique required that the data be spaced equally along the x-axis. The o r i g i n a l program was one provided by the Monroe calculator company, but was modified so that the best f i t data could be obtained from the c o e f f i c i e n t s derived from the c a l c u l a t i o n . In general, t h i s program worked very well but lacked the automated c a p a b i l i t y and s t a t i s t i c a l parameters provided by the multiple regression program. and D -b -m -r -e -169. 3. 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 T h i s program was w r i t t e n t o a l l o w d i r e c t comparison of data manually s e l e c t e d from the e q u i l i b r i u m scan to t h a t obtained from the data a c q u i s i t i o n system. The program produced absorbance v s . r a d i a l d i s t a n c e , absorbance v s . r a d i a l d i s t a n c e squared and'c(£)/c data s i m u l t a n e o u s l y , and s t o r e d the data i n t h r e e separate groups o f r e g i s t e r s . Any s e t of data c o u l d , t h e r e f o r e , be compared t o the a c q u i s i t i o n data by p l o t t i n g . The i n p u t and output o f t h i s program are presented i n F i g u r e 2A. D. FORTRAN Programs Two major FORTRAN programs were w r i t t e n ; the complete MWD program u s i n g m u l t i p l e r e g r e s s i o n , and the simplex o p t i m i z a t i o n r o u t i n e . Although the simplex r o u t i n e was i n c o r p o r a t e d i n t o the MWD program, i t was o r i g i n a l l y w r i t t e n as a separate program, so t h a t i t c o u l d be r e a d i l y m o d i f i e d f o r use i n other o p t i m i z a t i o n problems. In the f o l l o w i n g s e c t i o n the deck arrangement and p e r t i n e n t i n s t r u c t i o n s f o r running each program w i l l be presented. F i g u r e 3A i l l u s t r a t e s the deck order f o r the thr e e modes of the MWD c a l c u l a t i o n and f o r the simplex r o u t i n e . P e r t i n e n t i n f o r m a t i o n about the deck i s e x p l a i n e d below, i n r e f e r e n c e t o t h i s f i g u r e . Figure 2A. Input and output of the data conversion program for manually selected data. Input: A - number of data points - number of chart values between reference edges - c e l l bottom - meniscus - i n i t i a l concentration B - chart units from the meniscus to the c e l l bottom C - selected absorbance values Output: D - r a d i a l distance i n cm 2 E - x - In A F - ? - c(£)/c 171. 1 2 . 0 0 0 0 8 5 • c o o o 7 .9 • 4 0 0 0 6 3 < • 6 0 0 0 7 6 -- 4 0 0 0 2 8 5 « • 0 0 0 0 .7 « • 1 9 4 5 6 « > 8 9 7 1 1 5 - > 9 0 0 0 7 « • 1 3 8 1— 5 0 0 « 0 0 0 0 -7 « 1 1 9 4 4 0 3 -• 0 0 0 0 7 « - 1 0 0 7 3 3 3 « • 0 0 0 0 7 -•0 8 2 0 2 9 1« 0 0 0 0 7 • 0 6 3 2 2 3 9 -• C 0 0 0 B _ l C D 2 0 0 5 0 6 5 0 5 5 0 5 5 0 5 4 9 5 0 0 0 0 9 5 2 7 : 2 14 6 —I !•: 6 8 6 0 9 9 8 9 4 2 0 0 8 0 8 1 1 5 4 7 6 7 3 3 8 9 0 1 4 7 6 4 1 1 0 • 0 0 0 0 0 1 0 1 0 1 0 1 1 9 3 1 7 5 4 3 2 5 6 7 4 14 0 3 2 0 2 16 8 4 3 8 3 5 0 2 1 0 0 0 4 4 6 6 8 3 8 5 yxyxyxxxyxxyxyyxyyyyyxyxyyyxxxxxxxxxxxxxxxxxxxyxxxxxyxyxxxxxxxxxxxxxxxxxxxxxxxxx C J O B S E T U P FOR THE V A R I O U S MODES OF THE MHO C A L C U L A T I O N NOR M Ai—c -*trCtitr*-T-t ' ) N -^ 'OOE tSTUNON TO"" PASSWRD --j u i i t " n J.S3unp»K* o n j . S ! M u O ? |6»-KIJMAT u ?\ u ?. . n i) - - I5ono - -xono jtso v'.n 0 , o 0 3 n F - n a O . J S S S 2 , S S ; > 1 HfS' OF OA! A 0 ,003n F - n u — o ; n o o n - n-, n o n - t ~ " »EMlF u. l » S I G N ( I F K ~c — CARD t - C A P O - 8 -CARD 3 CARO a 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 of the MWD program and f o r the si m p l e x program. I T E R A T I V E HOOF J<!TGNP»« T O M M — - — - -SPUN noJ.PSUBPAK^OHJ.SIM^f)? (on-KlJMAT n o n o 0 , n 0 n 1 o o o n —evn-i r.o 3 n o o o - S M O O T H I N G - M O D E O . o o t n F - n i 0 . ? 5 S 3 RF S T OF T A T A o . w o < i i F - n u o . n n n o no 2 a " T E N D F T L E » 3 T G k J H F F c r -1STSM0N TO*» P A S S X O R O - — «RIIN o q J . S S l i 8 P » K * 0 B J . S I u w O ? 163-KIJMAT u ? I u 2 . 0 ? i s n n o - , ? S - A n n o ~ ? 5 o t ' . f l 6 . R o n ( 1 . 0 0 3 0 E - 0 1 1 0 . ? 5 S J ? , S 5 ? 1 R E S ' OF PA T A 7 . ? 0 0 . 2 « 3 7 3 u S f l E t l 7 , 7 0 0 . ? a j 7 S f l 5 8 E I I 0 , o i t « n F « n u O . o o o o F - n a 0 , » 0 3 0 E - o u I t « l J E ^ O F I L E • J S T G V O F F C •TTTnTVlVTf-o , n o o o o, nono 0 , n o o o o n i 0.?5n n . ? s n n . o o n " n o SiMBLFx P R O G R A M $ 3 T GW0N T P M P » S S * O B O ' J R U ^ np,j;si "Pir"* 8 I t 2 3 2 3 6 mono J E W " F I L E J S T G H O F F c 2 n 0 0 0 ^1 7 0 0 0 0 7UU anooo ? n o n o o 5 0 0 0 0 0 " C A R D " 1 C*RU 2 CARD 3 C A R D C A R O CARD 1 CARD 2 CARD 3 T A P O T CARD 5-CARD b CARD 7 CARD 8 CARD I CARD' ?. 173. 1. The molecular weight d i s t r i b u t i o n program a. Calculation mode Card 1 contains the following information: the number of multiples of the s t a r t i n g molecular weight for the series the number of data cards the number of cycles p - the molecular weight i n t e r v a l - the c a l c u l a t i o n mode indicator; 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 molecular weight of a series the power value of the molecular weight multiple the rotor speed the p a r t i a l s p e c i f i c volume the density of the solvent the c e l l meniscus the c e l l bottom RT 174. Only the f i r s t two terms were required for the c a l c u l a t i o n to proceed, with the rest being for reference only. Card 3: - X - K c ( U / c _ optics a l t e r n a t i v e , where 1 = UV optics, 2 = Schlieren Card 4: - A special n u l l set required at the end of each data set to force the intercept towards zero. b. Iterative mode The deck arrangement was s i m i l a r to the ca l c u l a t i o n mode except that i t contained an additional card. Card 1 changed i n meaning somewhat, with the f i r s t term now r e f e r r i n g to the number of components comprising the d i s t r i b u t i o n , and the t h i r d term becoming a constant that has to stay at a value of one. Card 2 was ac t u a l l y redundant, but was kept i n the deck. Card 5 was the i t e r a t i o n control card for the simplex optimization routine, 175. where the f i r s t term defined the maximum number of it e r a t i o n s , the second the number of components i n the system, the t h i r d the number of molecular weights associated with the components and the rest the range l i m i t s associated with the components. c. Smoothing mode Again the deck arrangement was si m i l a r to the c a l c u l a t i o n mode, except i t contained extra dummy cards that served to strengthen the frequency response obtained from the i t e r a t i v e c a l c u l a t i o n . Cards 4-6 were the dummy cards that contained the frequency values i n the p o s i t i o n found to be optimum by the i t e r a t i o n . Card 8 was the matrix card, with the f i r s t term defining the number of components i n the system, the second the number of terms in the matrix and the re s t the actual matrix values which had to be composed of zeroes and ones. This p a r t i c u l a r deck was set up for a three component system, the optimum frequencies of which were found to be 0.50, 0.25 and 0.25. The actual terminating matrix would have appeared as: 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 176. But was converted to a line a r vector: 100/100/010/010/0 01/001 A two component system was handled i n a simi l a r manner. 2. The simplex program The simplex algorithm was o r i g i n a l l y written as a separate program, designed to simulate the action of going through the multiple regression by picking up a vector of dummy F-rat i o values that were read i n . Thus, the program could be modified by removing the read statement and having i t cycle through whatever operation or cal c u l a t i o n which produces the response term to the factors. Card 1 defined the number of dummy response values along with the dummy values. Card 2 defined the number of factors, the number of range l i m i t terms and the range l i m i t p a i r s . After obtaining t h i s information, the program could follow through the algorithm picking up dummy response values and making the appropriate simplex decisions. 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N E . l ) GO TO 3 1 2 • ~~—~ : * R I T E < 6 , 3 3 3 > FORMA T ( • • 1 2 b ( I M * ) i / ) _GO Tn ? 7 3 1 2 4 3 CONTINUE - • — * R I T t ( 6 , 4 3 ) I I T S - ! 7 , ? _ l < , < * T - 3 8 , ' e N O 0 F C Y C L E NUMBER • • I 2 , / / , 1 X I 1 2 5 ( 1H * ) t / i c 2 7 CONTINUE ; — c c _ _ i _ _ _ N . I T T E R A T I O N ROUTINE U S I N G THE S I M P L E X ROUTINE c c I F I K L l K . f c O . O I G O TO 100 ENC = 0 1 0 0 N = N-1 ' — — IF' ( K L I K .fcQ, 1 ) GO TO 1 0 0 GO TO 110 CONTINUE IF (KOUNT ,GE, 1 ) GO TO 2 8 5 "~ " — c c S P E N D L A Y S C A L C U L A T I O N 2 0 9 » H I T E ( 6 , 2 0 9 ) : • K F _ A G » 0 T 1 , ' 1 ' , T * 9 ' , S l ' ' , P L E X , T E H A T , 0 N R O U T I N E ' • / I T 4 9 I 2 6 ( 1 H * ) I / / / ) R E A U . t b t l O l ) M P , K F . K M . ( M _ R ( J ) , J » 1 , K M ) 1 0 1 FUKMA1 ( j I b , 6 F ] o , 0 ) "—' »R IT E I b, 1 0 1 ) M P • KF tKM , ( Mn'R ( j ) , j o j ,KM ) A A = 1 , 0 / ( K F * ( S O R T ( 2 , 0 ) ) ) D = S()R'T ( KF + 1 . o ) O M - A A * ( H » l , 0 ) ""— •—— — P'1-AA* ( ( K F - 1 , 0 ) +B ) \ — C A L C U L A T E MATRIX CONSTANTS c K»l DO 210 J ' l , K M | 2 W T S ( K ) 1 J 1 2 1 0 K = K + 1 — K * l DO 211 J = 1 , KF RUM* ( J ) E H J R I K M ) - MwR < K ) 2 1 1 2 6 2 K = K + 2 — — F O R M A T ( 3 F 2 0 , 0 / / I c C A L C V L A TJL, .THE RANGE COVERED bY EACH MOLECULAR WEIGHT SET -c c c c GENERATE THE MATRIX I N THE FORM OF A L I N E A R VECTOR G E N E R A T I O N OF I M F lp. TFDMS c T O T » K F » ( K F + l ) K F R = K F + 1 J»l 0 0 2 1 2 K » K F R | T Q T t K F R y » T S ( K ) » R Q M W t J ) • P M + » T S . , | ) 2 1 2 J 3 J + 1 • C C G E N E R A T I O N O F A L U T H E IQI T E R M S K F E a K F + 2 K F N = K F - i K F E E = K F E + K F N __.2 0 0 2 B b N A " 1 i K F N D O 2 1 _ J = K F f c , K F E E * T S ( J ) a R O N i i k ( K ) * 0 M + » T S ( K ) 2 1 3 I F I K . E O . ( K F + 1 ) ) K = l C O N T I N L iE K F K L = K F t E + K F R K F L * K F f - + K F ' R 2 8 5 K * K + 1 C O N T I N U E I F l K F L A G , E 0 , 5) G O T O 2 4 0 | F ( K F L A G . f c O . 4) GO TO 2-17 I F ( K F L A G , E 0 . 3) G O T O 2 2 9 I F ( K F L A G , L 0 , 2 ) G O T U 2 2 6 I F < K F L A G . E O . 1 ) G O T O 2 1 7 - J J L l K F L A G . F:0 . f, ) T O p Q * . • i r ( K U A U , t o . 7 G o T o 2 9 7 ~ — — . E S T = 0 * O R S T = 1 0 0 0 0 0 0 N _ X T = 0 C C F I R S T F O U R R E G R E S S I O N S T O G E N E R A T E T H E K Z = K F F - T E S T S 2 1 8 K T - 0 J C = 1 K = 1 _ _ 0 2 1 b J B J C i K7 2 1 6 M » X ( K l l | B W T S ( J ) K = K + 1 K F L A f i s J -K.aU! iT = K ( ) U N T + 1 2 1 7 R E V I N 0 l b . GO T O 1 0 2 K T " K T + l »R I J F ( f, , 2 7 4 > 4 N S » ( 1 ( 1 ) o F T L S r ( K T ) = A N S W ( I O ) J C « J C + K F K Z = K _ + K F l* <-~T„ . L t . K F ) G O T O ->|n C C c c O R D E R - F I N D T H E O E S T A N D W O R S T F I T C A L C U L A T E S T H E L O C A T I O N O F E A C H F - T E S T I N R E L A T I O N T O T H E V E C T O R 2 S 1 D O 2 6 0 J = 1 , K F R I F ( F T f c S T ( J ) , L T , ( O E S T - 1 , 0 ) ) C O T O 2 8 0 B E S T = F T E S T ( J ) ._Qf_.ua j —> 2 8 0 C O N T INuE D U 2 1 9 J = 1 , K F R V ; I F I F T f c S T C J ) , G T . U O R S T + l , 0 t ) C-0 Tb it 1 9 — _ WORST « F T E S T ( J ) L O C K C J 219 CONTINUE K c c C A L C U L A T E AVERAGE VALUE F O R N E X T N t X T * 0 K = 1 00 _ 2 0 J = l , K f - ' P — — I F ( J , K 0 . LOCW) G O TO 2 2 0 I F ( J , E 0 , LOCH) GO T O 2 2 0 _ N E X r = F T K S T ( J l + N E X T -\ V ' #* 2 2 0 CONTINUE- ' — — — L N E X T - N E X T / < K F R - 2 ) ftHlTE(6.28l) P F S T . fcORSTi NEXT _ _ M _ J _ Q H !* A J_ L__Ii! E S 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 c C A L C U L A T E C E N T R O I D K P = K F K « « = L O C K I * K F — — — K H = K r KEND=T0T P2« C l K K | S | ) 00 2 2 3 K = K P » K F N D I K F " " — — 1 F ( K , E 0 . K » | GO T O 2 2 3 C I K X ) o * T S ( K ) + C ( K K ) 9 2 3 CONTINUE C ( K K ) =C < Kn ) / K F ' — KW o K w - l K K t K K - | K P - K D - 1 c r S t - l N U — l\ fc PJ U • 1 "—" — : I F ( K K , N t , 0) G O TO 2 2 4 c -CALCULATE T H F R F F I F C T m N c KW=LOCw*KF J - K F P 2 5 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 " — K F L A G a 2 GO TO 2'J9 2 g 6 > ± ' . ( _ A N S W ( 10) .GE. ( B E S T * i - n ) ) G O TO 2 2 7 c . G t . (NEXT + 1 , 0 ) ) G O T O 2 3 3 GO TO 2 3 8 ' 6 E ' " , 0 B S T + » • 0 1 1 <> 0 T O 2 3 3 O c c C A L C U L A T I O N O F E X P A N S I O N ~ ' — 2 2 7 R E F L E C - A N S w ( l O ) .K*=LOC**KF J*r.t- . 2 2 8 CONTINUE 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 2 2 8 K F L A G B 3 G O T O 2 9 9 ^ 2 2 9 I F ( A N S » ( 1 0 ) , G E . R E F L E C ) G O T O 2 3 1 ~C S U B S T I T U T E R E F L E C T I O N F O B W O R S T A N D C A L C N E « C E N T R 0 I 0 2 3 4 F T E S T ( L O C V ) a H E F L 6 C • v U R _ T = R E . F L _ C K ». = L 0 C * * K F 2 3 0 w I S < K W ) « r M » x ( J i 1 ) __.__!__. K 'ft = K A - 1 I F ( J . , 0 ) G O T O 2 3 0 GO TO 2 S 1 3 3 1 F T £ _ T t L P C * I ' A H S o ( 1 fl ) VfO"«"S~T ° A N S * I 1 0 ) c C S U B S T I T U T E E X P A N S I O N F O R W O R S T A N D C A U C N F * C E N T R O I D K'* = L O C * * K F J = Kf-2 3 2 I * T S ( > C I F I ) - M A X < J , 1 ) _________ I F ( J . N E . O ) G O T O 2 3 2 G O T O 2bl M _ _____ • £ I f E X P A N S I O N I S N O T A S G O O D A S R E F L f c C T I O N GO B A C K T O H E P L E C T I O N C 2 3 3 R F F L E C a A N S w ( 1 o ) —GQ--XQ__3_ C C A L C U L A T I O N F O R C O N T R A C T I O N O F T H E R E F L E C T I O N 2 3 5 K " - L Q .C ._J_Ki : WtFi_fcC = A N S * . io) J = K F 2 3 6 C O N T I N | j t ' _M»X ( J» l )• CCJ)+.5»<.ClJ)-WTS(Kvn). J » J - 1 I F ( J , N E , 0 ) G O T O 2 3 6 _ _ _ _ _ A _ _ L _ _ G O T O 2 9 9 C C C H E C K T O S E E I F C O N T R A C T I O N F I A L E D ? 3 7 I F ( A N b i V ( l Q ) , L E , ( R E F L E C + ) , 0 ) ) G O TO ? 9 A G O T O 2 9 8 ('"• C ^ C C A L C U L A T E C O N T R A C T I O N C L O S E S T T O T H E W O R S T L O C A T I O N C 2 3 8 K * O L 0 C * * K F R L F L E C * A N S * ( 1 o ) J 3 K F 2_U_-C_.NI.INU_ M * X ( j , i ) » C ( J)-,_»((C{J ) » W T S l K * • ) ) ) " K A = K A — 1 J S J - 1 J J L U . ^ E . O ) GO TP _ _ 9  K F L A G « _ G O T O 2 9 9 \ ' C c c I F C O N T R A C T I O N C L O S E R TO * 0 R S T F A I L S _ 2 A X L _ L F ( A N S J L U . 9 J . . . L E . C W O R S T - I . O M GO TO 2 9 5 2 9 8 F T E S T I L O C * ) • A N S * ( 1 0 ) — — " — W O « S T » F T E S T ILOCW) Kvv = L O C * * K F J « K F V • 241 C O N T I N U E — - — W T S ( K * ) • M W X ( J , 1 ) J 3 J - J K '.V = K n - 1 I F ( J . N f c . O ) GO TO 241 ~~ — — GO TO 251 c P A S S I V E H F F L F f r i n N C O N T R A C T I O N c 2 9 4 K* = LUC*'*KF J = K F 2 9 0 C O N T I N U E » « A i j , i ) - C ( J ) + a _ 5 M ( C ( J l . t t T S < K * ) > > — — j a j - l K w = K * " l J F ( J.Nf-.0 )6Q T O 290 — — GU TO 2 9 9 c M_L__i__E WORST r i l N T U A i - T i n N c 2 9 5 Kw»LOC**KF J=KF -2.9- CONT I -UE p»« * U , l | J I . , a b » U B S | (C<J>-*TS<KW) ) ) ) — K W « K * - 1 I f ( J . N f c . O ) GO TO 2 9 2 K F L A G = f o — GO TO 2 9 9 2 9 7 IF ( ANSw< 10) , L T , J R E F L f c C l ,0> ) GO TO 2 7 0 GO TO 2<ja 2Hb 1 r ( AN.* ( 1 0 I , L E . ( WORST- 1 . 0 > ) GO TO 2 7 0 ~ ~ ~ GO TO 2 9 B 2 7 0 * » l T r ( 6 , 2 7 l ) 2 _ _ L - £ O R ^ A T _ L / / / , T 4 5 > I0ME OF THF M A S S I V E C O N T R A C T I O N S F A I L F n • . / . T 4 S . i j a i l H ' l l . — GO TO 110 2 9 9 KOUNT=KOUNT+l 1 . F L K O U N T . G T . M D I « n TO I i n *R I Tr ( ft « 2 0 6 ) KOUNT — ' — " — — 2 8 6 FORMAT</. I I T T E R A T l O N N U M B E R ' | J 4 ) nf> I TE ( b , 2 7 4 1 ANSW ( 1 0 ) _ t _ A_F_ORM A r ( / I F - T E S T 1 . F ? n , -n r REMIND 16 — — — GO TO 102 w c • c c B E G I N I N G OF THE SMOOTHN I NG ROUT! N E ~ — — c se\? uiR I T K fi.QOO 1 9 9 9 FORMAT( 1 X , / , T 5 8 , • S M O O T H N I N G R O U T I N E ' i / i T 5 A . 1 8 ( 1H * ) • / ) " — R E A D ( - • 2 0 0 ) K A 1 K G , ( D U M M Y ( J ) • J » 1 t K G ) 2 0 0 F O H M M ( 2 I 3 . 4 0 F 3 . 0 ) f~) W R I T E ( 6 . 9 0 3 ) K A . K G . < D U M * < Y ( J ) | J B 1 | K G ) " ^ • C ^ ' v - S i l T ^ 0' D l , M W T V A R I A B L E S ' « I 3 1 2 X t K¥ = « N t l | » P * I R E M I N D 1 6 R F . A D < 1 6 . 1 6 > ( K I J M A T t K > t K - 1 i K V ) K B = 1 KI) = P - K A c c H-V.F. L O A P l - D K I . I M A T W I T H » H A T W A S OR I fi I N A l t y I N T H E M l F C KE=P+KD ft OO ____.f_ K F = K E J = l DO 3 0 0 K » 1 i K C _ _ _ _ _ O 0 _ K i _ J _ i j _ A . K I J M A 1 ( K F ) - O U M M V ( J ) K F = K F t 1 . " J t 1 __)j_XQNiaj____ K F = K F - K A KF = K F t P 3 0 0 C O N T I N U E C C D O L O O P 3 0 0 K E E P S T R A C K O F T H E N U M B O F K I J P E R C / C O K B = K f i + 1 K E = K F t P I F ( K B , G T . I ) G O T O 2 0 4 G O i n 4 0 p 2 0 4 C O N T I N U E R E w1 N O l b * R I T f c ( l 6 H 6 ) ( K l J M A T ( K ) , K " l f K Y ) G O T O 2 J _ C C C S C H L I E R E N O P T I C S R O U T I N E ( F O R M U L A C H A N G E ) W H E N D A T A ( « t l ) « 0 7 0 0 C O N T I N U E AJL A L « i i-^ -jL' 1 • K P , * * 2 , * ( * W X ( J . K ) * * 2 ) * D E X P ( _ 0 A T A ( 1 , K P ) * M W X ( J , K > « GO T O 7 0 1 1 1 0 S T O P E N D S U B R O U T I N E D A T A R t T U R N E N D L i s t i n g of the portion of the S c i e n t i f i c  Subroutine Package used for the MWD program C DECK _£ c c c J: C O R R ,CORR S U B R O U T I N E C O R R E P U R P O S E — = — 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 O F 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 , C O R R E < N , M . I O , X , X B A R , S T O , R X , R , _ , D , T ) U S A G E C A L L D E S C R I P T I O N OF PARAMETERS N - NUMHFH OF O B S E R V A T I O N S , N MUST M m NUMBER OF V A R I A B L E S , M r.^UST BE IO - O P T I O N CODE FOR INPUT DATA J2 IF DATA ARE TO QE READ IN FR0 M H E > > O R O R B TO 1 , 2. CORR CORR -CORR CORR CORR CORR __CQRR CORR CORR CORR CORR S P E C I A L S U B R O U T I N E NAMED DATA, USED BY T H I S S U B R O U T I N E BE L C"• ) 1 IF A L L DATA ARE ALREADY IN CORE, XE—LOJLQJ) I _ _ V_AL.UE _OF_X J 5_0 , 0_» TO INPUT D E V I C E I N ( S E E S U B R O U T I N E S I F 10=1, X B A R S T D OUTPUT OUTPUT x i s D A T A , V E C T O R _.f c i O R 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 R X D E V I A T I O N S . - O U T P U T M A T R I X OF OF L E N G T H L E N G T H C O N T A I N I N G C P N T A I N I N G M E A N S i S T A N D A R D (M X M) C O N T A I N I N G SUMS OF C R O S S -P R O D U C T S OF D E V I A T I O N S FROM MEANS, 0J2IPUI_MATRix [ONLY UPPER TRIANGULAR P O R T I O N OF THE SYMMETRIC MATRIX OF »• llv M ) c 0 N T A I NTNG CO R~R E L A T I O N C O E F F I C I E N T S . ( S T O R A G E MQDfc OF 1) OUTPUT VECTOR OF L E N G T H M C O N T A I N I N G THE DIAGONAL OF THE u A„lPJ.X_gX^UJlS_OF__C_R Q_S S - PRODUCTS OF  D E V I A T I O N S F R O M M E A N S . D - W O R K I N G V E C T O R O F L E N G T H M T - W O R K I N G V E C T O R O F L E N G T H M R E M A R K S C O R R E H I L L NOT A C C E P T A CONSTANT V E C T O R , SU^RUUTINFS ANG F .UAX.IJLQ--.S-.. RH.OS RA M.S__R LQ UJLP E12 O A T A ( M ^ O ) - T H I S S U B R O U T I N E M U S T M E P R O V I D E D B Y T H E U S E R , ( l ) I F i o = 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 T O 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 j £ _ i E R N A i _ J L N P J J T D E V J C E CORR CORR CORS THECORR CORR CORR CORR _CORR CORR CORR CORR _CORR CORR CORR CORR _CORR CORR CORR CORR _CORR_ CORR CORR CORR _C.QRR. CORR CORR CORR _£ORR_ ( 2 ) M E T H O D P R O D U C T - M O M E N T CORR CORR AN CORR , CORR IF 1 0 = 1 , T H I S S U B R O U T I N E I S NOT USED BY CORR CORRE BUT MUST E X I S T IN J O - DECK. IF USER CORR HAS NOT S U P P L I E D A S U B R O U T I N E NAMED DATA t CORR THE FO IJ-.Q "IMS I S S U G G E S T E D . C ORR S U B R O U T I N E DATA CORR RETURN CORR END CORR - CORR CORR CORR CORR 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 , 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.QR EL_: CORR CORR 0 0~ 30 40 50 60 70 SO _ ?o I 00 1 10 120 _130 1 40 150 160 170 ""180 190 200 210 ""220 230 240 2 5 0 260 270 260 29 0 "3 00 3 t 0 320 JJO 340 350 360 380 3 9 0 400 4J0 4 2 0" 430 440 _5 0_ 460 470 480 490 500 510 520 bJQ 540 550 560 5J.Q 580 590 U l MEN. I ON 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 C If- A O 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 I S D E S I R E D . C C I N C O L U M N i S H O U L D B E R E M O V E D F R O M T H E R O U B L E P R E C I S I O N C S T A T E M E N T . V H I C H F O L L O W S . D O U B L E P R E C I S I O N X H A H . S T n . R X . R i B . T i X  CTJPTT C O R R I I M I . C O R R .CO PR. T H E C O R R C O R R C O R R 6 0 0 6 1 0 6 2 0 _.-l.Q_ 6 4 0 bbO 6 6 0 THE C MUST A L S O BE A P P f c A R I N G I N OTHER ROUT I N E , R E M O V E D F R O M 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 U S E D I N C O N J U N C T I O N W I T H T H I S T H E 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 M U S T A L S O _ C D - N J A I N D Q I J B J L E 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 . S O R T A N D A B S I N S T A T E M E N T 2 2 0 M U S T B E C H A N G E D T O 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 D O 100 j a 1 F M .3.LJ ...= 0.0 1 0 0 T ( J | « 0 , 0 K « ( M * K + M ) / 2 D O 102 I = l t K 1 0 2 R I I I ' O i l l F N C N L » 0 L - 1 1 0 ) 105 t 1 2 7 t 108 D A T A A R E A L R E A D Y I N C O R E 105 D0_a_a_._____l__-I = 1 ,N D O 107 107 T ( J ) = T ( J ) + X ( L ) XB A R I J ) s T 1 J ) 108 T ( J ) = T ( J ) / F N D O 1 1 5 1 = 1 . N JK = 0 L ° I - N D O 1 1 0 _L_r.L.t__ D ( J ) aX ( I. )-T ( J ) J 1 0 B ( J ) =L} ( J ) +0 ( J ) 0 0 115 J a 1 , M 0 0 l i b K M . J J K a J K + I l i b R ( J K ) = H ( J K )+0(J)*0(K) G O T O 2 0 5 R E A D O A S E R V A T I O N S A N D C A L C U L A T E T E M P O R A R Y M E A N S F R O M T H E S E D A T A I N T U J 1 27 130 I F . N . M ) 1 3 0 . 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A T i n N S F R O M R E G R F R H ^ n N S S D R = 0 ( L 1 l - S S A R V A R I A N C E O F E S T I M A T F •  F N = N - K - \ S V C S S O H / F N S T A N D A R D D E V I A T I O N S O F R E G R E S S I O N C O E F F I C I E N T S 0 0 . 1 3 0 J = l i K L l * K J J _ J j j _ L t _ l _ L » I S A V E ( J I 1 2 5 S 8 ( j ) B O S O R T f D A B S ( ( R X ( L 1 ) / D ( L > I * S Y ) ) C O M P U T E D T - V A l U E S  J 3 0 T < J ) = B ( J ) / S B ( J > S T A N D A R D E R R O R n F E S T I M A T E J 3 5 S Y * D S Q M T ( D A B S ( S Y ) ) F V A L U E  F K = k S S A R M a s S A R / F K -S.S U R_M = £ S D R /F N F " S S A R M / S S O R M A N S I I ) = B O A N S ( 3 ) = S Y A N S ( u ) > S S A R ANS( f t 1=F K _AN S ( t> ) a S S A R_M_ A N S 1 7 ) = SSi-R A N S ( 8 ) = F N A N S ( 9 ) = S S D R M _ANSJ 1 Q ) a F R E T U R N E N D DECK M I N V M U L T 1 1 6 0 M U L T 1 1 7 0 M U L T 1 1 8 0 _ j M U J _ . T _ l . i 9 0_ M U L T 1 2 0 0 M U L T 1 2 I 0 M U L T 1 2 2 0 M U L T 1 2 3 0 M U L T 1 . 4 0 M U L T 1 2 5 0 M U L T 1 2 6 0 -_MUL.T_1. _ / _ 0 _ M U L T 1 2 8 0 M U L T 1 2 9 0 M U L T 1 3 0 0 ____JL.T m o _ M U L T 1 3 2 0 M U L T 1 3 3 0 M U L T 1 3 4 0 _ _ _ _ L I ! . _ _ 0 _ M U L T 1 3 6 0 M U L T 1 3 7 0 M U L T 1 3 8 0 - J - . U L I J.3 9 0 _ M U L T 1 4 0 0 M U L T 1 4 1 0 M U L T 1 4 2 0 — M U L T 1 4 3 0 _ M U L T 1 4 4 0 M U L T 1 4 5 0 M U L T 1 4 6 0 M U L T 1 4 7 0 _ M U L T 1 4 8 0 M U L T 1 4 y 0 M U L T 1 5 0 0 M U L T 1 5 1 0__ M U L T 1 5 2 0 M U L T 1 5 3 0 M U L T 1 5 4 0 _ M U L T 1 5 5 0 M U L T 1 5 6 0 M U L T 1 5 7 0 M U L T 1 5 8 0 _ M U L . T 1 5 9 Q _ _ M U L T 1 6 Q 0 M U L T 1 6 1 0 M U L T 1 6 2 0 _ t _ U L I . l 6 3 0__ M U L T 1 6 4 0 M U L T 1 6 5 0 M U L T 1 6 6 0 ____. I l b 7 0 _ M U L T 1 6 8 0 M U L T 1 6 9 0 M U L f 1 7 0 0 _-__Jt-.T_L7_l___ M U L T 1 7 2 0 M U L T 1 7 3 0 -J___JiY 1 0 i M I N V M I N V 20 30 VD S U B R O U T I N E MINV P U R P O S E I N V E R T A MATRIX MINV MINV M INV TV 50 60 70 U S A G E C A L L M I N V ( 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 M A T R I X , D E S T R O Y E D IN C O M P U T A T I O N AND R E P L A C E D BY R E S U L T A N T I N V E R S E , N - f lRDFR OF M A T R I X A  0 - R E S U L T A N T DETFRMJNANT :  L - WORK. M • *'ORK VECTOR VECTOR OF OF L E N G T H L E N G T H REMARKS MATRI X A MUST UE A GENERAL MATRIX S U B R O U T I N E S AND F U N C T I O N SUBPROGRAMS R E Q U I R E D NONE METHOD __—THE... STANDARD G A U S S - J O R D A N METHOD IS U S E D . THE DETERMINANT I S ALSO C A L C U L A T E D , A D E TERMINANT OF ZERO I N D I C A T E S THAT THE MATRIX IS S I N G U L A R , S U B R O U T I N E M I N V ( A » N , 0 • L » M ) D I M E N S I O N A( l ) , L ( 1 I,M( t ) I F A C 1N DOUBLE COLUMN S T A T E M E N T WHICH FOLLOWS. D O U B L E P R E C I S I O N A i D • b I G A tHOLD 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 IS D F S I R f c D , THE 1 SHOULD BE REMOVED FROM THF DOUBLE P R E C I S I O N THE C MUST A L S O BE A P P E A R I N G IN OTHER R O U T I N E , 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 USED IN C O N J U N C T I O N WITH T H I S THE D OUBLE 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 ALSO C O N T A I N DOUBLE P R E C I S I O N FORTRAN F U N C T I O N S , ABS IN S T A T E M E N T 10 MUST BE CHANGED TO DABS, S E A R C H F O R L A R G E S T E L E M E N T 0 = 1 .0 N K » - N DO ao K » 1 « N _NK55 N K t__ L ( K ) = K » ( K ) SK K K « N K + K B I G A = A ( X K • DO IZ< 20 J i N* i J -K »N 1 ) MINV 80 MINV 90 MINV 100 _ _ _ . _ - _ 3 _ _ _ J U _ MINV 120 MINV 1 JO MINV 140 _ _ I N V _ 1 5 0 MINV 160 MINV 170 MINV 180 MINV 1 9 0 MINV 2 0 0 MINV 2 10 MINV 2 2 0 _MIN_V_ 23 0 MINV ~2 4 6 MINV 250 MINV 2 6 0 _MINV_ 2 70 MINV 280 MINV 2 9 0 MINV 3 0 0 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 _MI._LV_ MINV MINV M INV __. I NV_ M I N V M INV M INV M INV 4 1 0 __J0 4 4 0 450 4 6 0 _470 4 80 490 500 5 10 , MINV 5 2 0 MINV 5 3 0 MINV 5 4 0 _MINV__50 MINV 5 b 0 MINV MINV -MJ_Ny_ MINV MINV MINV MINV MINV MINV 67 0 5 8 0 5 9 0 6 00 61 0 6 2 0 _____ 6 4 0 6 b 0 0 0 20 I « K . N I J = l Z + l 10 I F ( D A B S ( Q I Q A ) i I S B I G A a A . I J ) ' D A B S ( A ( I J ) ) ) 1 5 * 2 0 . 2 0 20 L ( K ) Ml C O N T I N U E C c c c 25 30 1N.ERCHANGE ROWS J = L I K ) . I F ( J - K ) 3 5 . 3 5 . 2 5  K I = K-N DO 3 0 1 = 1 i N K I =K1+N HOID=-A { K l | J I -K. I «K + J A ( K 1 ) 5 . ( J 1 ) A ( J I ) "HOLD I N T E R C H A N G E COLUMNS 35 3B I-If. ( K ) I F ( I - K ) 4 5 . 4 5 . 3 H JP=N*( I - 1 ) OU 4 0 J = 1 i N JK=MK+J - L L i L J P j t J 40 H U L D * - A ( J K ) A( JK ) = A ( J I ) A ( J I ) =HOLD D I V I D E COLUMN BY MINUS P I V O T ( V A L U E OF P I V O T ELEMENT I S C O N T A I N E D IN B I G A ) 45 I F ( B I G A ) 40.4ft.aB 46 0=0,0 RE TURN 46 00 65 1 = 1 § N IL± 1 -J____5j___LSu 50 IK=NK+I A( I K ) = A ( IK ) / ( . 55 CONTINUE" J3_CL. .BIGA) RfcUUCE MATRIX DO b5 I • 1 • N IK =NK t I HUL D * A( I K ) I J = I - N DO 65 J = 1 « N -XJ__I.__tN_ I F ( I - K ) bO , 6 5 . 6 0 60 I F ( J - K ) 6 2 , 6 5 . 6 2 62 K J M j - l + K A( I J ) = H O L D * A I K J ) + A ( I J ) 65 C O N T I N U E D I V I D E ROW BY P I V O T K J « K - N DO 75 J > l i N MINV 6 6 0 MINV 6 7 0 MINV 6 8 0 MINV 6 9 0 700 7 1 0 720 _7JQ_ 74 0 750 76 0 _7 7 0_ 7 8 0 790 800 _8 i o_ MINV B 20 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 NV_8 9 0_ MINV 9 0 0 MINV MINV M I NV M I NV MINV MINV MINV 910 9 2 0 930_ 94 0 95 0 9 6 0 9 7 0 9 8 0" 99 0 MINV M INV M I N V 1 0 0 0 _M I NV 1 0 1 CL M I N V 1 0 2 0 M I N V 1 0 J 0 MINV104 0 __M J N V 1 0 5 0_ M I N V 1 0 6 0 M I N V 1 0 7 0 M I N V 1 0 8 0 M I N V I 0 9 0_ M I N V 1 10 0 M I N V 1 1 1 0 M I N V 1 1 2 0 M I N V I 1 3 0_ M I N V . 1 4 0 M I N V 1 1 5 0 M I N V t 1 6 0 _M_INV 1 1 7 0_ M I N V 1 1 8 0 M I N V 1 1 9 0 M I N v 1 2 0 0 J_i.N.yj2io_ M I N V 1 2 2 0 M I N V 1 2 3 0 M I N V 1 2 4 0 -JVI.N.V_12 5_L M I N V 1 2 6 0 M I N V 1 2 7 0 H1 VD 7 P , T C , . M I N V 1 « ! 8 0 7 0 A C K j T - i . K j ! ^ , ' ? ? M I N V 1 2 9 0 ™ ) K J ' , K J I / B ' " M I N V 1 3 0 0 c f n 1 u t . — — — Mj.Ny l J 1 P_ C P R O D U C T O F P I V O T S M } N V ! J ! § c n s n j B i f . M I N V I 3 A O ^ P-°*Hl<>* . M^ V13bO... C R E P L A C E P I V O T U V R E C I P R O C A L M I N V I I W O BO C O N T I N U E = J • MINV1 A 00 MINV1 A 10 C F I N A L «0V» A N D C O L U M N I N T E R C H A N G E M I N V 1 4 2 0 £ M.I Ny 1 43 0_ K » N MI N V 1 4 4 0 1 00 K = ( K- 1 ) M I NV I 4i>0 1 F ( K ) 1 5 0 f I b O t 1 0 5 • M I N V l A b O t 0 5 _ IJ»LXKJ H.I N V. 1 4 7 0_ I F ( I - K ) I 2 0 i l 2 0 » 1 0 6 M I N V 1 4 8 0 JOB J U » N » H » ] ! M I N V J 4 9 0 J R c N * ( I - 1 ) M l N V l b O O DCL-.1.1 Q_J-M i N f* I NV. 1 = 1 Q_ JK = j ( j + j M1NV1&20 H O L D = A ( J K ) M I N V 1 S 3 0 J I = J R + J • M I N V 1 5 4 0 ft_( JK. )=--&( J I ) M J N.v 1 6 b 0_ 110 A ( J I > = H O L D M I N V 1 5 6 0 120 J = M ( K ) MINVlt>70 I F ( J - K ) 1 0 0 » 1 0 0 t l 2 b M I N V 1 5 8 0 D O 13 0 I = 1,N ~ — B.I.N.V-1 b 9 0_ K7 = KV + N, M I N V l f . 0 0 J . L 5 K J - J S ± J _ . _ _ _ _ " N V 6 2 0 K K i i s . i / I I I •— — — •. MJ..N .VL1 a J V_ 1 3 0 ilSi ZDW KJKXltSg 60 10 100 M I N V l b 5 0 , 5 0 RETURN M I N V 1 6 6 0 ^ C ^ - ^ — — M1_NV_16J^_ M I N V 1 6 B 0 2 01. G. L i s t i n g o f t h e s i m p l e x p r o g r a m 202. 0> LT I o at tn Z < — r_ a o » x m -)ui| 2 IS) X r- < a * z * O -z - -I o — x| — o a ui — r 2 — U l » -2 U l 4|> — JJ UJ i o . - a • z a _ i _> u _ o at or — _ a _ — a -» n — -m i t- » • a - 5 3-. o « r u r_ — i>-• ru » o _ . — I- J . L l O — —at < e r a _j * u. at a z as 3-H« • I P -ruiru _ , — o _ - ! _ || < i u l •-2 f ! r - Z a l — _> c ; a c U! - _ 2 S -t _£ S.-H s o -•in o — — ru o z j - 3 - 4 4 _|o -4 a: a — — o U . U _ •-n > 1X1 o at u. _ u. exj — — _t ra o o — mi • — -— o — +" • I X — X • _ J _ : I i t _ ui d a * * « 3 4 4 — —»u) ii n * 4 1 r _ o a o * — _ . a o _td — O a • o ru II u. i _ — — — ru 4 — + ru * + z ui _: — r- 0 « D i O H O * _ a j a ic x o 01 . — r u — to r u m 3 a — 4 4 a a ui U) X U.I -t -tl UIO OOUllJU i z h a U l O x 13 h x i a +• u. z _: a • » h — o -> • a 3 L L O _: a n _r n ru — ru — +1 . ui T| B a ->a « - i ui z a ui ui X _ l _1 4 Z o z ni — u i -fj I a. x x _ _q _ II _ U l s o lui » -I |U1 z X o a H -J U n x z xi ru x x n c i|-> -— -> • rw — ui o O _ r J _ j . - O I O * a l -ii u _ c — a ( U l o — Z h — 4 h Z z a o o u x I n * — 1 _ njru K F E f c « K F E f c + K F R /~\ K F E c K F E + K F R K » K - f 1 gab C 0 (SI I IJ ME . I F (K f UAG ,fcO, b) GO 10 240 IF ( K F L A G .EQ, 41 GO TO 2 3 T IF I K F L A G , E 0 , 3 ) GO TO 2 2 9 IF (KHL.AG . F Q . 2) GO TO ? 2 b : I F ( K F L A G .EO. 1) GO 10 2 1 7 I F ( K F L A G . E O . 6 ) GO TO 2 9 9 I F ( K F L A G , E Q . 7 ) G O TO 29*, l i E J i i . g 0 i O W S T = 1 0 0 0 0 0 0 0 0 — N E. X T = 0 C C F I R S T FOLIO. KF GR E S S I ON 5 TO G E N E R A T E THE F - T F S T S  C K P = K F K I =0 : : 2ia K = I DO 3 1 b J = J C i KP KFLA6=1 fiQ_r.o_i.oi : 2 1 6 « » X ( K i | ) = A T S ( J > 2 l 7 K T = K T t l F T t S T U T ) » ANSW (10) J X g j l C t K F : K P a K P + KF I F (KT , L E , K F ) GO TO 2 1 8 C C ORDLR- F I N D T M F OfcST AND WORST F I T  C 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 IN RtLAT ION TO THE VECTOR C g b l DO 2 8 0 J = 1 » K F R . L t l F I F- S T ( J ) . L T . B E S T ) GO TO 280 ;  BEST =F T E S T < J > LOCfci = J 280 C O N T I N U E B.Q 2 19 J = l .KFR ; l F ( F T t S T < J ) , 0 T , WORST) GO TO 219 »iORSI » F T E S T ( J > L O C * " J 2 1 9 C O N T I N U E -C c C A L C U L A T E A V E R A G E V A L U E 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 .EO, LOCO) GO T O 2 2 0 N E X T = F T E S T ( J l + M F X T ro •J 2 2 0 C U N T I N U E N F : X T = ^ E X T / ( K F R - 2 ) * R I T F ( 6 i 2 7 4 ) ( F T F S T ( J ) •J=1 ,4) * B l T t ( n i 2 8 1 l flF«iT«*np«-T i M F X T o CO • - c 2 0 1 F U R i i A T ( « r i E S r i , 2 X i F l b . 5 , i * 0 R S T - « 2 X , F l b , b » I N E X T . | 2 X | F 1 5 , 5 ) C c C A L C U L A T E C E N T R 0 1 D K P = K F K w a L O C i i K * K F A 204. M i - x l o o t- II »> « ~ I \ I | OX.M| z < UJ — O « ftl ft! »»> ftl ftl — X. o x — X * — rxtt •- X • * * O i l — ul a D a z n X X *- ¥ - K Z J : — o — — a o a n ftl fti o ftl U-— •o •»t»i • ft! « X ft: —fti n • c ->-• J , — 3 o - > — 1 3 — C a a — — i -o z ~ oi • 10 o >!z fti • » - X , 0 -i x i x —>-x a II x x o u n z ' x. a ui UJ «3 — — £ u. a o x x x x l — * u. u c X T (0 — X a * a it ui — — 3 — -O ui — 2 ftl I •!! — -> — H f V3 >- — X I Z X * T( o i s n o z o l -o M ftl X - > < 1 II — J . < u. u. : x — x MM] ftl C i o t -I-o o | 1 3 K X </>Ul U J Z I 1-13 : s 3 » M ——I ^ O — i » « i | / | •nz _ — < — I I O U i L . o — O — JO M ftl ftl z o <t — u. o — v3 Jt z « u. ui • 3-o u a l o _ l j u j i I u.»iizx-i Ul * U I O t H a x - » o z ~> I M ftl I Z I rx|-» < U — -x i -: ro ft) o _l U. Ul a x j o -J «IO - I I I -< a u - a : x * » M o Ui _ l u. — ui — a • -> B — U X — Ul * 3 -1U. O U . X a 3 L l * a _ i a * — ro ftl a o — —) • « . , I UJ ftl — M O ! < < z t - r - d X — X| " O " ai l / 1 J i - I tjor uivm -> tti l — 3 *! B — M c U_ O Xj " J =c "» xj — 1 3 ftl 1 ftl ( (' ' ( ( 2 3 1 F T t S T I L O C w ) « A N S * 1 1 0 > " — — — K O H S T B A N S V . ' ( 1 0 ) c S O O S T I T U T E E X P A N S T O M F O P W O R S T AMD C A L C N E W C W N T R n i D c K V < = L O C * » K F J = K F W T S ( K * ) = M w x < J . M j a j - 1 — — ;  K * * K w ™ 1 I F ( J . N E . O ) G O T O 2 3 2 G O T O ? « . ! c c I F E X P A N S I O N I S N O T A S G O O D A S R E F L E C T I O N _ 0 BACK T O R E G L E C T I O N _ 2 _ 3 R E F L F C B A N S * ( i n ] G O T O 2 3 4 ' — c c C A L C U L A T I O N F O R C O N T R A C T I O N O F T H E R E F L E C T I O N 2 3 5 2 3 b K * = L O C » * K F — — — R E F L t C = A N S w I 1 0 ) J » K F C O N T I N I P E M * X ( j i l ) » P ( J ) + , 5 * ( 1 P I J ) . » T S ( K * I ) ) — J = J - 1 I F t J . N E . O I G O T O 236 1 2/a K r - L A G - 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 G O ro 102 c c C H E C K T U S E E I F C O N T R A C T I O N F I A L E D I F ( A N S V v I i o ) , l . T , R E F L E C ) G O T O 2 9 4 G O T O ? 9 8 c c C A L C U L A T E C O N T R A C T I O N C L O S E S T T O T H E W O R S T L O C A T I O N ? . 8 K'A = L 0 C * * K F 2 3 9 R E F L E C « : A N S * ( i o > " — — J 3 K F C O N T l N u E 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 ) G O T O 2 3 9 K F L A _ = H o c 2 7 9 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 O N - » . 1 , 2 X » 3 F 2 0 , 0 ) G O T O 1 0 2 c I F C O N T R A C T I O N C L O S E R T O W O R S T FATLS ~~ 8 * 0 ? 9 f l I F ( A N S w ( 1 0 ) , L T , * O R S T ) G O T O 2 9 5 F T E S T l L O C * ) a A N S M 1 0 ) » O R S T « F T f c S T ( L C V C » ) K * • L O C « i * K F 206. O V9 O — • j> I — J J I M • — • o X -> i -II — s u a X i i i j 2 3 2 O Z o 3 ac i z — c — OJ X > -( -—in ~< — 5 Cf -u o J U . J I I X J u » CM z c < cr z in ac cd 3 z x «x! I I t - > X J J J J J £ ~i -> > I z ii — —! -< '•3 J J < c m i 4 i - i h in - _ l ~ c r < u . u . at 3 a r U J < 3 u z 0 •-• J U . 1 -I I X z 1 n o 4-- • i n J J I l i t '* • * ->< n x ~ x i J t * I M O ftf x I Z — o ~>-t • I -—u i < I —z — a x > s —« s tt — i n — 4 Ot-(V • • X JJ— IV - r - U OJ < O i < — -x _1 ac a a. * u . x »» CP ft: o — 5) — «| q —0" — a • H «<\ i * ni I in OIZ c z ct i <* •- < g i J J a> cr o U l tn z o z a t_i tn •a Z o H + • OJ t -• z ? I O 0 3 — — o ) — — X ' I- u I <t 3 t -• £ t - Z • or 3 3 O O j u. ts x o — at i ^ r - o run. »4 JJ • o ftl u. t a. * Z U l 3 Z o -x • O — X — 1 - J I — tn • Ui J I • xzj-0 i - M 3 -ivs z »a~ 3 - x! in — o •—- »-a x z t - i i 1 3 u i — - cd-> t - J J x f-1 4 O X - H l r -• fti • r u " > »- • \H • r*. Z J J " " " ^ j 1 * 1 ™" o u i < —I u i a x i - z i f - — a — - . n t n — o o l |u- a 323ct o t - z ) * u . <1 < t3 W iii ) ; 

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